UQLIB
USER MANUAL
M. Moustapha, C. Lataniotis, P. Wiederkehr, P.-R. Wagner, D. Wicaksono, S.
Marelli, B. Sudret
CHAIR OF RISK, SAFETY AND UNCERTAINTY QUANTIFICATION
STEFANO-FRANSCINI-PLATZ 5
CH-8093 Z
¨
URICH
Risk, Safety &
Uncertainty Quantification
How to cite UQLAB
S. Marelli, and B. Sudret, UQLab: A framework for uncertainty quantification in Matlab, Proc. 2nd Int. Conf. on
Vulnerability, Risk Analysis and Management (ICVRAM2014), Liverpool, United Kingdom, 2014, 2554-2563.
How to cite this manual
M. Moustapha, C. Lataniotis, P. Wiederkehr, P.-R. Wagner, D. Wicaksono, S. Marelli, B. Sudret, UQLIB – User
manual, Report UQLab-V2.1-201, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Switzerland,
2024
BibTeX entry
@TechReport{UQdoc_21_201,
author = {Moustapha, M. and Lataniotis, C. and Wiederkehr, P. and and Wagner, P.-R.
and Wicaksono, D. and Marelli, S. and Sudret, B.},
title = {{UQLib user manual}},
institution = {Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich,
Switzerland},
year = {2024},
note = {Report UQLab-V2.1-201}
}
Document Data Sheet
Document Ref. UQLAB-V2.1-201
Title: UQLIB – User manual
Authors: M. Moustapha, C. Lataniotis, P. Wiederkehr, P.-R. Wagner, D. Wicak-
sono, S. Marelli, B. Sudret
Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich,
Switzerland
Date: 15/04/2024
Doc. Version Date Comments
V2.1 15/04/2024 UQLAB V2.1 release
V2.0 01/02/2022 UQLAB V2.0 release
V1.4 01/02/2021 UQLAB V1.4 release
• Added the uq map function
V1.3 19/09/2019 UQLAB V1.3 release
• Added the Graphics library
V1.2 22/02/2019 First release of UQLIB
Abstract
UQLIB is a collection of general-purpose open-source MATLAB libraries that are useful in the
context of uncertainty quantification. These functions are currently used across the scientific
modules of UQLAB, but they are designed for generic use.
This user manual serves as a reference documentation for all the relevant functions of UQLIB.
The manual includes the algorithm and explanation behind each library, its syntax, input and
output, and at least one example demonstrating its usage.
In the current release, UQLIB includes the following libraries:
• Differentiation
• Optimization
• Kernels
• Input/output processing
• Graphics
Keywords: UQLAB, Differentiation, Optimization, Kernels, Input/output processing, Graph-
ics
Contents
Contents 3
Introduction 1
Organization of the library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Organization of the function documentation . . . . . . . . . . . . . . . . . . . 3
How to read the input/output table . . . . . . . . . . . . . . . . . . . . . . . . 3
Notes on usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Differentiation 7
uq gradient – First-order numerical differentiation 7
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Optimization 14
uq gso – Grid-search optimization 14
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
uq ceo – Cross-entropy optimization 19
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
uq cmaes – Covariance Matrix Adaptation Evolution Strategy (CMA-ES) 26
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
uq 1p1cmaes – (1+1)-CMAES 35
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
uq c1p1cmaes – Constrained (1+1)-CMAES 43
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Kernel 52
uq eval Kernel – Compute kernel matrix 52
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Input/output processing 58
uq subsample random – Random subsampling 58
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
uq subsample kmeans – k-means clustering-based subsampling 60
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Graphics 63
uq figure – Create a figure 63
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
uq formatDefaultAxes – Default formatting of an Axes object 66
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
uq plot – Create a formatted 2D line plot 69
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
uq bar – Create a bar plot 72
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
uq histogram – Create a histogram 75
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
uq violinplot – Create violin plots 80
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
uq
scatterDensity – Create a scatter plot matrix 84
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
uq traceplot – Create trace plots 88
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
uq legend – Create a legend 91
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Functional programming 94
uq map – Map a sequence using a function 94
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
References 110
Introduction
UQLIB is a collection of general-purpose open-source MATLAB functions that are useful in
computational science and engineering, particularly in the context of uncertainty quantifi-
cation (UQ). The functions were originally created during the development of the UQLAB
scientific modules, but they are designed to be usable standalone.
UQLIB libraries cover a wide range of computational goals, from optimization to efficient
kernel evaluation. In contrast to most other UQLAB user manuals, this manual is not intended
to be an introduction to the theory behind a particular problem and its possible solutions. It
is conceived instead as a detailed reference guide to deploy the provided functions outside of
the UQLAB environment.
1 Organization of the library
UQLIB is organized into different independent libraries. The functions within a library share
similar computational goals or objectives. The following sections summarize each of the
libraries.
1.1 Differentiation library
Differentiation of mathematical functions is related to efficiently compute gradients. The
UQLIB function uq_gradient approximates the first-order derivative (gradient) of a multi-
dimensional function at multiple points.
1.2 Optimization library
Optimization is the process of finding the minimum or maximum of a multi-dimensional
function. In general, optimization algorithms can be split into local and global optimizers.
The former relies on local information, e.g., gradients, to iteratively solve the optimization
problem, while the latter has a larger scope in exploring the entire search space.
The optimization library of UQLIB comprises global optimization algorithms for the solution
of continuous single-objective problems. The following algorithms currently are available:
• Grid-search optimization (uq_gso)
• Cross-entropy optimization (uq_ceo)
1
UQLIB
• Covariance matrix adaptation–evolution strategy (CMA-ES) optimization (uq_cmaes)
• (1+1)-Covariance matrix adaptation–evolution strategy ((1+1)-CMA-ES) optimization
(uq_1p1cmaes)
• Constrained (1+1)-Covariance matrix adaptation–evolution strategy (Constrained (1+1)-
CMA-ES) optimization (uq_c1p1cmaes)
Each of these implementations can handle bound constraints, but only the variant of CMA-ES
(i.e., the last algorithm) can handle non-linear constraints.
1.3 Kernel library
Multi-dimensional kernel functions are useful in a variety of applications such as function
interpolation, Gaussian process modeling, representation of random fields, etc. An arbitrary
function is generally not a valid kernel as it has to fulfill the so-called Mercer’s conditions
(Cherkassky and Mulier, 2007). Furthermore, kernel functions also feature some parameters
that shall be tuned according to a particular application.
The function uq_eval_Kernel computes the kernel matrix of two input matrices for a speci-
fied kernel function. The library supports popular stationary and non-stationary kernel func-
tions, as well as custom user-defined kernels.
1.4 Input/Output processing
UQLIB includes miscellaneous functions to assist in the processing of the input and output of
an uncertainty quantification analysis tasks using UQLab.
1.4.1 Subsampling
Consider a large sample set X =
x
(1)
, . . . , x
(N)
where each sample point is an M -
dimensional vector. In certain contexts, such as metamodeling (Marelli et al., 2021; Latanio-
tis et al., 2021), having a large number of sample points N leads to high-computational costs
or even renders the calculation intractable. Subsampling refers to the process of reducing the
number of sample points in a way that some of their statistical properties are retained.
In UQLIB, the functions uq_subsample_random and uq_subsample_kmeans create a sub-
sample from a full sample set based on simple random sampling and k-means clustering,
respectively.
1.5 Graphics library
This library comprises functions that facilitate the creation of plots with a unified look by
allowing to set defaults for selected properties (e.g., font size, grids, color order). Many
of the plotting functions wrap the respective MATLAB function and sets the UQLAB default
formatting style. Consequently, all input arguments valid for the corresponding MATLAB
function are also valid for these functions.
UQLAB-V2.1-201 - 2 -
User manual
2 Organization of the function documentation
This user manual contains a concise reference for each UQLIB library, as well as at least one
example for each function that demonstrates its usage.
Each function in the library is documented according to the following structure:
• Objective briefly states the purpose of the function;
• Algorithm or Description presents the algorithm (if any) or the description of the
function as well as provides important references;
• Syntax lists all the different possible function calls, followed by brief description for
each of the different calls;
• Examples gives at least one application of the function showing its input and output;
• Input provides exhaustive lists of inputs of the function, including their names, data
types, dimension, and short descriptions;
• Output provides exhaustive list of outputs from the function, including their names,
data types, dimensions, and short descriptions;
• Notes gives additional important details and remarks about the function if any, ranging
from further detail on the implementation to possible dependencies. If there is no
additional remark, this section is excluded.
The input and output sections are presented using a series of tables. The instruction on how
to read such tables are given in the following section.
3 How to read the input/output table
A series of tables are used to describe all the inputs and outputs of a given function. Each
table commonly contains the name, data types, dimensions (when applicable), and short
descriptions.
3.1 Input table
The main inputs of a function is presented in a 4-column table illustrated below, for a function
called uq_foo having several input arguments: input1, input2, options, and Name-Value
pairs.
Table 1: uq_foo(input1, input2, options, Name, Value)
input1 N × M Double First input.
input2 Double Second input.
Continued on next page
UQLAB-V2.1-201 - 3 -
UQLIB
Table 1–continued from previous page
options Structure, see Table 2 Additional options of the function, as
structure.
Name, Value Name-value pairs, see
Table 4
Additional options of the function, as
name-value pairs.
The first column in the above table indicates whether a given input argument is mandatory,
optional, mutually exclusive, etc. A comprehensive list of the symbols and their meaning are
given in the following table:
Mandatory
Optional
⊕ Mandatory, mutually exclusive (only one of
the fields can be set)
Optional, mutually exclusive (one of them
can be set, if at least one of the group is set,
otherwise none is necessary)
The other three columns in Table 1 correspond to the name, data type, and description of the
input argument. When applicable, the dimension of an input argument is given explicitly.
3.2 Structure inputs and outputs
MATLAB structures play an important role in the user interface of UQLAB and therefore
UQLIB. They offer a natural way to semantically group configuration options and output
quantities. All the field names of a given input or output structure are listed in a separate
3-column table. This is illustrated below for the options structure appeared in Table 1.
Table 2: uq_foo(..., OPTIONS)
.Field1 String
default:
'default_string'
Description of Field1.
.Field2 Double
default: 0.5
Description of Field2.
.Field3 Logical
default: false
Description of Field3.
.Field4 Structure, see Table 3 Description of Field4.
The first column in the above table corresponds to the name of the field. Notice that a field
of a structure can be identified by the dot notation, i.e., the name is prefixed by a period. The
second column corresponds to the data type of the field. When applicable, the dimension and
UQLAB-V2.1-201 - 4 -
User manual
the default value are also given. Finally, the last column corresponds to the short description
of the field.
Due to the complexity of the algorithms implemented, it is not uncommon to employ nested
structures to fine tune inputs or present more complex outputs. In that case, the fields for
each nested input/output structure are elaborated using another 3-column table illustrated
below for the .Field4 structure in Table 2.
Table 3: options.Field4
.NestedField1 Double Description of NestedField1.
.NestedField2 Integer Description of NestedField2.
3.3 Name-value pair inputs
Another approach to pass options to a function is by specifying the so-called name-value
pairs. Using this approach, an optional argument is passed to a function by specifying the
name of the argument as a string and followed immediately by the value for that particular
argument. Several UQLIB functions use name-value pairs to specify optional arguments. All
the available argument names and the corresponding valid values of a function are listed in
a separate 3-column table as illustrated below.
Table 4: uq_foo(..., NAME, VALUE)
'NamedArgument1' String
default: 'Value1'
Description of the argument
'NamedArgument1'.
'Value1' Description of the value 'Value1'.
'Value2' Description of the value 'Value2'.
'Value3' Description of the value 'Value3'.
'NamedArgument2' Integer
default: 2
Description of the argument
'NamedArgument2'.
The first column in the above table corresponds to the names of the arguments. Notice that
a named argument is always specified as a string. If only a limited selection of values of an
argument is possible, these values are listed in the second column of the table as illustrated
above for the named argument 'NamedArgument1'.
3.4 Output table
Finally, UQLIB functions often results in more than a single output. All the outputs of a
function are presented in a table similar to the ones shown previously and now illustrated in
Table 5. When applicable, the dimension of an output is given in the second column of the
table.
UQLAB-V2.1-201 - 5 -
UQLIB User manual
Table 5: [output1,output2,output3] = uq_foo(...)
output1 Vector Double Description for output1.
output2 Matrix Integer Description for output2.
output3 Structure Description for output3.
As mentioned in Section 3.2, structures can become outputs of a function. They can also
be further nested. The documentation for such outputs is given in separate tables similar to
Table 2 and Table 3.
4 Notes on usage
All functions of UQLIB automatically becomes available in the current MATLAB environment
upon the launch of UQLAB. These functions, as other UQLAB functions, begin with the prefix
uq_. Help can be accessed from within MATLAB using either the command help or doc
followed by the name of the function.
Most of the UQLIB functions are, however, self-contained and can be used independently
from UQLAB. Dependencies, if any, are noted in the respective Notes section of each function
documentation.
The source code for the UQLIB functions are available in the lib folder inside the main
UQLAB installation folder. The subfolders within the lib folder are organized according to
UQLIB libraries. To get the exact location of a given function within these subfolders, use the
command which followed by the name of the function in the MATLAB command window.
- 6 - UQLAB-V2.1-201
uq gradient – First-order numerical
differentiation
1 Objective
Compute the gradient of a multi-dimensional function at given points.
2 Algorithm
The gradient of a multi-dimensional scalar-valued function f(x) = f (x
1
, x
2
, . . . , x
M
) is a
vector that consists of the partial first-order derivatives of f(x) with respect to each dimen-
sion:
∇f =
∂f
∂x
1
,
∂f
∂x
2
, . . . ,
∂f
∂x
i
, . . . ,
∂f
∂x
M
T
. (1)
2.1 Finite difference
The basis of numerical approximation for derivatives is Taylor expansion. The function f
is expanded using a Taylor expansion for x
i
+ h while keeping the other dimensions x
∼i
constant
f (x
i
+ h, x
∼i
) = f (x) + hf
x
i
+
1
2
h
2
f
x
i
,x
i
+ . . . = f(x) + hf
x
i
+ O(h
2
), (2)
where h is the so-called step size, f
x
i
is the first-order derivative with respect to dimension x
i
,
O is the higher-order terms, and O(h
2
) indicates that the lowest order of these terms is 2. By
neglecting higher-order terms and solving for f
x
i
yields the finite difference approximation of
the derivative:
f
x
i
≈
f (x
i
+ h, x
∼i
) − f(x)
h
. (3)
In particular, the above formulation is called the forward difference approximation. Truncat-
ing the higher order terms in Eq. (3) results in a truncation error of order 1.
Eq. (3) can be reformulated if the function f in Eq. (2) is expanded for x
i
−h. In other words,
f (x
i
− h, x
∼i
) = f (x) − hf
x
i
+
1
2
h
2
f
x
i
,x
i
+ . . . = f(x) − hf
x
i
+ O(h
2
). (4)
Following the same procedure results in the backward difference approximation of the deriva-
7
UQLIB User manual
tive:
f
x
i
≈
f(x) − f (x
i
+ h, x
∼i
)
h
, (5)
in which the order of the truncation error remains 1.
The third approximation results from combining Eq. (4) and Eq. (2). Rearranging and solving
for f
x
i
results in the centered difference approximation of the derivative:
f
x
i
≈
f (x
i
+ h/2, x
∼i
) − f (x
i
− h/2, x
∼i
)
h
. (6)
In this formulation, the order of the truncation error is 2, hence it is more accurate. However,
it requires one extra function evaluation per input dimension with respect to the forward
and backward differences. Details on the derivation as well as error analysis can be found in
Chapra and Canale (2015).
2.2 Methods
uq_gradient offers all three methods to approximate the gradient of a function at a given
point. The cost of the approximation in terms of the function evaluations N
T
is N
T
= N ×
(M + 1) for the forward and backward methods and N
T
= N ×2M for the centered method;
where N and M are the number of points and input dimensions, respectively.
Figure 1 illustrates the approximation of the gradient for the function f (x) = sin (x) at
x = (2.4π, sin(2.4π))
T
by the three methods, assuming a fixed step size of h = 0.5. As can be
seen the resulting gradient depends on the method.
2.3 Step size h
Choosing the proper value for the step size h is important for numerical accuracy. A large
value of h can result in a worse gradient approximation (Figure 1). On the other hand,
a very small value of the step size can result in a very small difference between f(x) and
f (x
i
± h, x
∼i
) that can numerically be indiscernible due to the finite precision of floating
point operations. In other words, the so-called round-off error will start to dominate the
approximation. By default, uq_gradient uses a fixed h = 10
−3
for each dimension.
2.4 Vector-valued function
uq_gradient supports functions with multiple outputs (i.e., vector-valued functions). In this
case, the approximation of the gradient is carried out for each output separately.
3 Syntax
G = uq_gradient(X, FUN)
G = uq_gradient(X, FUN, GradientMethod)
G = uq_gradient(X, FUN, GradientMethod, FDStep)
G = uq_gradient(X, FUN, GradientMethod, FDStep, GivenH)
G = uq_gradient(X, FUN, GradientMethod, FDStep, GivenH, KnownX)
G = uq_gradient(X, FUN, GradientMethod, FDStep, GivenH, KnownX,...
- 8 - uq gradient UQLAB-V2.1-201
UQLIB – Differentiation
Figure 1: The three methods to estimate the gradient of f = sin(x) around x
0
= 2.4π
with step size h = 0.5. The forward method approximates the gradient using P
0
= f(x)
and P
f
= f (x
0
+ h) (blue dashed line). The backward method approximates the gradient
using P
0
and P
b
= f(x
0
− h) (red dash-dot line). Finally, the centered method uses P
−
c
=
f(x
0
−0.5h), P
+
c
= f(x
0
+ 0.5h) (green dotted line). The black solid line is the true gradient.
Marginals)
[G,M_X] = uq_gradient(...)
[G,M_X,Cost] = uq_gradient(...)
[G,M_X,Cost,ExpDesign] = uq_gradient(...)
G = uq_gradient(X, FUN) returns the gradient G of the function FUN evaluated at the
points X given as (N × M ) matrix, where N is the number of points and M is the
number of input dimensions. It uses the 'forward' method and a step size of 10
−3
.
G = uq_gradient(X, FUN, GradientMethod) uses the approximation method specified
in GradientMethod (see Table 1).
G = uq_gradient(X, FUN, GradientMethod, FDStep) allows selecting the type of the
step size type by specifying FDStep (see Table 1).
G = uq_gradient(X, FUN, GradientMethod, FDStep, GivenH) allows for adjusting the
step size by specifying GivenH. The specific effect of GivenH on the step size depends
on the selected FDStep (see Table 1).
G = uq_gradient(X, FUN, GradientMethod, FDStep, GivenH, KnownX) uses KnownX,
a set of precalculated values of FUN at X, instead of evaluating the function on X within
the code. If KnownX is provided, the cost is reduced by N.
G = uq_gradient(X, FUN, GradientMethod, FDStep, GivenH, KnownX, Marginals)
UQLAB-V2.1-201 uq gradient - 9 -
UQLIB User manual
uses the standard deviations of input dimensions stored in the structure Marginals.
Marginals is part of a UQLAB INPUT object.
[G,M_X] = uq_gradient(...) additionally returns the values of FUN at the points X.
[G,M_X,Cost] = uq_gradient(...) additionally returns the Cost of the approximation
in terms of the total number of function evaluations.
[G,M_X,Cost,ExpDesign] = uq_gradient(...) additionally returns a 1 × N structure
array containing the experimental designs used in the approximation of the gradient
vector at each given point in X.
4 Examples
4.1 Approximate the gradient at different points
Approximate the gradient vector of the function:
f (x) = 5 + 2x
2
1
+ 3x
3
2
(7)
at the points x
(1)
= (3, 0.5) and x
(2)
= (0.5, 1). The analytical solution for the gradient at
those points are:
∇f
|x
=
∇f
|x
(1)
T
∇f
|x
(2)
T
=
12 2.25
2 9
The following code approximates the gradient vectors with minimum number of inputs given
by the user:
fun = @(X) 5 + 2
*
X(:,1).ˆ2 + 3
*
X(:,2).ˆ3;
X = [3 0.5; 0.5 1];
G = uq_gradient(X,fun)
The code produces:
G =
12.0020 2.2545
2.0020 9.0090
in which each row of the output is the gradient vector approximation at a given point.
4.2 Approximate the gradient of a vector-valued function
Approximate the gradient vector of the vector-valued function:
f(x) =
x
3
1
+ x
2
2
,
2
3
x
3/2
2
, 25 + 0.5x
1
+ 10x
2
, x
1
x
2
2
T
at the points x
(1)
= (3, 4) and x
(2)
= (3.5, 9). The analytical solution for the gradient at those
- 10 - uq gradient UQLAB-V2.1-201
UQLIB – Differentiation
points are defined per output component:
∇f
1|x
=
∇f
1|x
(1)
T
∇f
1|x
(2)
T
=
27 8
36.75 18
∇f
2|x
=
∇f
2|x
(1)
T
∇f
2|x
(2)
T
=
0 2
0 3
∇f
3|x
=
∇f
3|x
(1)
T
∇f
3|x
(2)
T
=
0.5 10
0.5 10
∇f
4|x
=
∇f
4|x
(1)
T
∇f
4|x
(2)
T
=
16 24
81 63
The following code approximates the gradient vectors with minimum number of inputs pro-
vided by users:
fun = @(X) [X(:,1).ˆ3+X(:,2).ˆ2 2/3.
*
X(:,2).ˆ(3/2) ...
25 + 0.5
*
X(:,1) + 10
*
X(:,2) X(:,1).
*
(X:,2).ˆ2];
X = [3 4;3.5 9];
G = uq_gradient(X,fun)
The code produces an N × M × N
out
multi-dimensional array, where N = 2, M = 2, and
N
out
= 4 are the numbers of input points, input dimensions, and output dimensions, respec-
tively:
G(:,:,1) =
27.0090 8.0010
36.7605 18.0010
G(:,:,2) =
0 2.0001
0 3.0001
G(:,:,3) =
0.5000 10.0000
0.5000 10.0000
G(:,:,4) =
16.0000 24.0030
81.0000 63.0035
UQLAB-V2.1-201 uq gradient - 11 -
UQLIB User manual
5 Input
Table 1: uq_gradient(X, FUN, GradientMethod, FDStep, GivenH, KnownX, Marginals)
X N × M Double Points at which to approximate the
gradient.
FUN 1 × N
out
Function handle Vector-valued function for which the
gradient is approximated.
GradientMethod String or function handle
default: 'forward'
Method for the gradient
approximation.
'forward' Use the forward method.
'backward' Use the backward method.
'centered' Use the centered method.
Function handle Use a user-specified function handle
to evaluate the gradient on X. The
custom function only takes X as
input.
FDStep String
default: 'fixed'
Specifies the step type.
'fixed' Use step size h =GivenH×1.
'relative' Use step size h =GivenH×σ
i
in the
direction of X
i
. Only available if
Marginals (see below) is provided.
GivenH Double
default: 0.001
“Step-size ratio”. Used to compute
the step size, the effect depends on
FDStep (see above).
KnownX N × N
out
Double Allows user to provide precalculated
evaluations of FUN(X).
Marginals Structure Marginals structure of a UQLAB
INPUT object. It is used to read the
standard deviations if FDStep is set
to 'relative'.
6 Output
Table 2: [G,M_X,Cost,ExpDesign] = uq_gradient(...)
G N × M × N
out
Double Gradient approximations at points X for
all outputs of FUN.
M_X N × N
out
Double Function evaluations FUN(X). Equal to
KnownX if provided.
Continued on next page
- 12 - uq gradient UQLAB-V2.1-201
UQLIB – Optimization
Table 2–continued from previous page
Cost Scalar Double Total number of function evaluations
done by uq_gradient.
ExpDesign 1 × N Structure Array Experimental designs built at each point
in X. Each structure contains the field:
• X, a Matrix Double of input
points.
• Y, a Vector Double of
corresponding function values.
7 Notes
• For probabilistic input, uq_gradient offers the possibility of using relative step size. In
this case, the standard deviations of the inputs are multiplied by h to obtain the step
size for each input. Probabilistic inputs are passed into uq_gradient via a structure
(see Marginals in Table 1).
UQLAB-V2.1-201 uq gradient - 13 -
uq gso – Grid-search optimization
1 Objective
Solve the following unconstrained optimization problem:
x
∗
= arg min
x∈D
X
f (x) , (1)
where x ∈ D
X
⊆ R
M
is an M-dimensional vector; D
X
=
Q
M
i=1
x
lb
i
, x
ub
i
represents the
search space, with the lower and upper bounds of the i-th input dimension x
lb
i
and x
ub
i
,
respectively; x
∗
is the optimal solution; and f is a scalar-valued objective function.
2 Algorithm
Grid-search optimization is a heuristic algorithm which consists in finding the minimizer of
an objective function among a predefined set of candidates. The algorithm is often used
for the calibration of hyperparameters in the context of machine learning applications. The
basic idea is to generate a grid over the input space, evaluate the objective function on the
generated points, and select the minimizer in this set as the approximate solution.
The procedure is as follows:
1. Initialize the algorithm:
• Define a set of candidate points to be evaluated or simply set the bounds of the
search space.
• Define the options for the optimizer such as the number of discretization points
per input dimension d
i
≥ 2, i = 1, . . . , M.
2. If a grid is not defined, create one:
• Generate a uniform discretization along each input dimension: X
i
=
n
x
(1)
i
, x
(2)
i
, . . . , x
(d
i
)
i
o
,
where x
(1)
i
= x
lb
i
and x
(d
i
)
i
= x
ub
i
.
• Generate the grid by tensorization: X =
Q
M
i=1
X
i
.
3. Evaluate the objective function at the grid points f
x
(i)
, i = 1, . . . , N, where N is the
size of the grid.
14
UQLIB – Optimization
4. Rank (sort) the grid points in increasing order of the objective function values, i.e.,
x
(1)
, . . . , x
(N)
defined such that f
x
(1)
≤ . . . ≤ f
x
(N)
.
5. Return the approximate solution: x
∗
= x
(1)
.
3 Syntax
XSTAR = uq_gso(FUN, MYGRID, NVARS)
XSTAR = uq_gso(FUN, MYGRID, NVARS, LB, UB)
XSTAR = uq_gso(FUN, [], NVARS, LB, UB)
XSTAR = uq_gso(FUN, MYGRID, NVARS, LB, UB, OPTIONS)
[XSTAR,FSTAR] = uq_gso(...)
[XSTAR,FSTAR,EXITFLAG] = uq_gso(...)
[XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_gso(...)
XSTAR = uq_gso(FUN, MYGRID, NVARS) finds a local minimizer of the function FUN using
only evaluations at a predefined set of points MYGRID. NVARS is the input dimension
(number of design variables) of FUN.
XSTAR = uq_gso(FUN, MYGRID, NVARS, LB, UB) finds a local minimizer of the function
FUN using only evaluations at a predefined set of points MYGRID, and only evaluating
data points that are within lower and upper bounds defined by LB and UB, respectively.
XSTAR = uq_gso(FUN, [], NVARS, LB, UB) finds a local minimizer of the function FUN
using on an automatically generated grid with 5 discretization points along each input
dimension, within the lower (LB) and upper (UB) bounds.
XSTAR = uq_gso(FUN, MYGRID, LB, UB, OPTIONS) finds a local minimizer of the func-
tion FUN using only evaluations at a predefined set of points MYGRID, with the default
optimization options replaced by the values in the OPTIONS structure (see Table 2).
[XSTAR,FSTAR] = uq_gso(...) additionally returns the value of the objective function at
the solution XSTAR.
[XSTAR,FSTAR,EXITFLAG] = uq_gso(...) additionally returns an exit flag that indicates
the exit condition of the algorithm, either an optimal solution is found or all specified
points fall outside the bounds (see Table 3).
[XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_gso(...) returns an additional structure with
information about the optimization process (see Table 3).
4 Examples
4.1 Minimize Rosenbrock’s function
Consider the minimization problem of the Rosenbrock’s function:
x
∗
= arg min
x∈[−10,10]
2
100(x
2
− x
2
1
)
2
+ (1 − x
1
)
2
. (2)
UQLAB-V2.1-201 uq gso - 15 -
UQLIB User manual
The minimum of this function is located at x
∗
= (1, 1) with the minimum value f
∗
= 0.
The following code solves the optimization problem using uq_gso on an automatically gen-
erated grid with the default 5 discretization points along each input dimension (see Table 5):
fun = @(X) 100 .
*
(X(:,2) - X(:,1).ˆ2).ˆ2 + (1 - X(:,1)).ˆ2;
nvars = 2;
lb = [-10 -10];
ub = [10 10];
xstar = uq_gso(fun, [], nvars , lb , ub)
The code produces:
Local minimum found that satisfies the bound constraints.
obj. value = 1
ans =
0 0
This solution differs from the analytical solution, but it is the best one found inside the grid.
4.2 Specify the number of discretization points
By default, the number of discretization points along each input dimension is 5. The total
number of points in the generated grid is thus 25. The following code can be used to increase
the size of the grid:
fun = @(X) 100 .
*
(X(:,2) - X(:,1).ˆ2).ˆ2 + (1 - X(:,1)).ˆ2;
nvars = 2;
lb = [-10 -10];
ub = [10 10];
Options.DiscPoints = 30;
xstar = uq_gso(fun,[], nvars, lb, ub, Options)
The code produces:
Local minimum found that satisfies the bound constraints.
obj. value = 0.128437
xstar =
1.0345 1.0345
With 30 discretization points along each input dimension, the total number of points in the
generated grid is 900.
- 16 - uq gso UQLAB-V2.1-201
UQLIB – Optimization
5 Input
Table 1: uq_gso(FUN, MYGRID, NVARS, LB, UB, OPTIONS)
FUN Function handle Objective function to be minimized.
MYGRID N × M Double Candidate set for searching the
solution.
NVARS INTEGER Number of variables in the objective
function to be optimized (M).
LB Scalar or 1 × M Double
default: -Inf
Lower bounds of the search space.
UB Scalar or 1 × M Double
default: Inf
Upper bounds of the search space.
OPTIONS Structure, see Table 2 Algorithm-specific options.
Table 2: uq_gso(..., OPTIONS)
.Display String
default: 'final'
Level of output display.
'none' Displays no output.
'iter' Displays output at each iteration.
'final' Displays only the final output.
.isVectorized Logical
default: true
Specifies whether the objective function
is vectorized.
true Objective function is vectorized.
false Objective function is not vectorized.
.DiscPoints Scalar or 1 × M Double
default: 5
Number of discretization points:
• The given value must be larger
than 1.
• When the problem is multi-
dimensional and a scalar is given,
the value is replicated along all
input dimensions.
UQLAB-V2.1-201 uq gso - 17 -
UQLIB User manual
6 Output
Table 3: [XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_gso(...)
XSTAR 1 × M Double Optimal solution.
FSTAR Double Objective function value at the optimal
solution.
EXITFLAG Integer Flag indicating the termination
condition of the algorithm
1 Approximate solution found.
-1 None of the user-specified grid points
belong to the bounds.
OUTPUT Structure, see Table 4 Diverse information about the
optimization process.
Table 4: [...,OUTPUT] = uq_gso(...)
.message String Exit message.
.funccount Integer Total number of objective function
evaluations.
.History Structure, See Table 5 History of all grid points and their
corresponding objective function values.
Table 5: OUTPUT.History
.Grid M × N Double Grid points that have been evaluated in
the search for an optimum.
.Fitness N × 1 Double Objective function values corresponding
to the evaluated points.
- 18 - uq gso UQLAB-V2.1-201
uq ceo – Cross-entropy optimization
1 Objective
Solve the following unconstrained optimization problem:
x
∗
= arg min
x∈D
X
f (x) , (1)
where x ∈ D
X
⊆ R is an M-dimensional vector; D
X
=
Q
M
i=1
x
lb
i
, x
ub
i
represents the search
space, with the lower and upper bounds of the i-th input dimension x
lb
i
and x
ub
i
, respectively;
x
∗
is the optimal solution; and f is a scalar-valued objective function.
2 Algorithm
The cross-entropy method was originally developed by Rubinstein (1997) for the estimation
of the probability of rare events. The method has been adapted by Rubinstein and Davidson
(1999) for the solution of continuous and combinatorial optimization problems and consists
in sampling iteratively the search space using a parametrized random distribution to converge
to the optimal solution. The implementation in UQLIB considers a Gaussian distribution for
sampling the candidate solutions.
The algorithm is summarized below following Kroese et al. (2006):
1. Initialize the algorithm:
• Set the parameters of the initial Gaussian distribution: the mean µ
[0]
x
and the
standard deviation σ
[0]
x
, which correspond to the starting point and initial global
step size of the algorithm, respectively.
• Set the internal parameters of the algorithm: the number of points per iteration
(or generation) N
pop
; the smoothing parameters α
CE
, β
CE
, and q
CE
; and the num-
ber of points in the elite sample set of size N
el
= bρ · N
pop
c, where b◦c denotes
the floor function and ρ is a coefficient such that 0 < ρ < 1. The elite sample set
corresponds to a subset of the best points with respect to their objective function
values.
• Set t = 1, where t is the counter for the algorithm iteration.
2. Sample N
pop
points
x
(1)
, x
(2)
, . . . , x
(N
pop
)
from a truncated Gaussian distribution
19
UQLIB User manual
N
[
x
lb
,x
ub
]
µ
[t−1]
x
,
σ
[t−1]
x
T
· I
N
pop
· σ
[t−1]
x
, where I
N
pop
denotes an identity matrix of
size N
pop
× N
pop
.
3. Evaluate the objective function on the sample points f
x
(i)
, i = 1, . . . , N
pop
.
4. Select N
el
sample points with the smallest value of the objective function. Those sample
points are denoted by
x
∗(1)
, . . . , x
∗(N
el
)
.
5. Compute the variable-wise mean and standard deviation of the elite sample:
e
µ
[t]
x
=
n
eµ
[t]
x
1
, . . . , eµ
[t]
x
M
o
, eµ
[t]
x
m
=
1
N
el
N
el
X
j=1
x
∗(j)
m
e
σ
[t]
x
=
n
eσ
[t]
x
1
, . . . , eσ
[t]
x
M
o
, eσ
[t]
x
m
=
v
u
u
t
1
N
el
N
el
X
j=1
x
∗(j)
m
− eµ
[t]
x
m
2
.
(2)
where x
∗(j)
m
is the m-th component of the j-th elite sample point.
6. Update the parameters of the Gaussian distribution:
µ
[t]
x
= α
CE
e
µ
[t−1]
x
+ (1 − α
CE
)x
best
,
σ
[t]
x
= β
CE
t
e
σ
[t−1]
x
+ (1 − β
CE
t
)x
best
,
(3)
where x
best
is the best solution found so far and β
CE
t
is defined as:
β
CE
t
= β
CE
+ β
CE
1 −
1
t
q
CE
(4)
7. If convergence is achieved, stop the algorithm; otherwise increase t ← t + 1 and go to
Step 2. The following convergence criteria are considered:
• Maximum number of generations: the algorithm stops if the number of genera-
tions (i.e., iterations) reaches a given threshold.
• Number of stall generations: the algorithm stops if the number of successive iter-
ations without sampling a point that improves the current best solution reaches a
given threshold.
• Number of function evaluations: the algorithm stops if the number of calls to the
objective function reaches a given threshold.
• Stagnation of the objective function: the algorithm stops if the absolute difference
between the maximum and minimum of the objective function values over a given
number of iterations (i.e., its range) is below a given threshold.
• Stagnation of the solution: the algorithm stops if the possible change in the so-
lution becomes extremely small, i.e., the current Gaussian distribution can only
sample points that are extremely close to its mean.
• Minimum values: the algorithm stops if the value of the objective function falls
- 20 - uq ceo UQLAB-V2.1-201
UQLIB – Optimization
below a given threshold.
The algorithm stops when any of these criteria is reached.
3 Syntax
XSTAR = uq_ceo(FUN, X0, SIGMA0)
XSTAR = uq_ceo(FUN, X0, SIGMA0, LB, UB)
XSTAR = uq_ceo(FUN, X0, SIGMA0, LB, UB, OPTIONS)
[XSTAR,FSTAR] = uq_ceo(...)
[XSTAR,FSTAR,EXITFLAG] = uq_ceo(...)
[XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_ceo(...)
XSTAR = uq_ceo(FUN, X0, SIGMA0) finds a local minimizer of the function FUN with X0
as the starting point and SIGMA0 as the initial variable-wise standard deviation.
XSTAR = uq_ceo(FUN, X0, SIGMA0, LB, UB) defines a set of lower and upper bounds
such that LB(i) <= XSTAR(i) <= UB(i) . If LB and UB are finite and X0 = [] and/or
SIGMA0 = [], the center of the search space (LB(i)+UB(i))/2 and 1/6 of the search
space width, i.e., (UB(i)-LB(i))/6 are used as X0(i) and SIGMA0(i), respectively.
XSTAR = uq_ceo(FUN, X0, SIGMA0, LB, UB, OPTIONS) minimizes with the default op-
timization options replaced by the values in the OPTIONS structure (see Table 2).
[XSTAR,FSTAR] = uq_ceo(...) returns the value of the objective function at the solution
XSTAR.
[XSTAR,FSTAR,EXITFLAG] = uq_ceo(...) returns an exit flag that indicates the termina-
tion condition of the algorithm (see Table 3).
[XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_ceo(...) returns a structure with additional
information about the optimization process (see Table 3).
4 Examples
4.1 Minimize Rosenbrock’s function
Consider the minimization problem of the Rosenbrock’s function:
x
∗
= arg min
x∈[−10,10]
2
100(x
2
− x
2
1
)
2
+ (1 − x
1
)
2
. (5)
The minimum of this function is located at x
∗
= (1, 1) with the minimum value f
∗
= 0.
The following code solves the optimization problem using uq_ceo by assuming default values
for the starting point X0 and initial global step size SIGMA0 (see Table 5):
rng(100, 'twister') % Fo r r e p r o d u c i b l e r e s u l t s
fun = @(X) 100 .
*
(X(:,2) - X(:,1).ˆ2).ˆ2 + (1 - X(:,1)).ˆ2;
lb = [-10 -10];
ub = [10 10];
UQLAB-V2.1-201 uq ceo - 21 -
UQLIB User manual
xstar = uq_ceo(fun, [], [], lb, ub)
The code produces:
Maximum number of stall generations (options.nStall) reached
obj. value = 0.0111671
xstar =
0.9071 0.8279
4.2 Specify the starting point and initial global step size
By default, the starting point of uq_ceo is X0(i) = (LB(i)+UB(i))/2 while the initial
global step size is SIGMA0(i) = (UB(i)-LB(i))/6. The following code specifies user-given
values for these two parameters:
rng(100, 'twister') % Fo r r e p r o d u c i b l e r e s u l t s
fun = @(X) 100 .
*
(X(:,2) - X(:,1).ˆ2).ˆ2 + (1 - X(:,1)).ˆ2;
lb = [-10 -10];
ub = [10 10];
x0 = [-5 5];
sigma0 = [2 2];
xstar = uq_ceo(fun, x0, sigma0, lb, ub)
which produces:
Possible change in X is below options.TolX
obj. value = 5.5885e-24
xstar =
1.0000 1.0000
5 Input
Table 1: uq_ceo(FUN, X0, SIGMA0, LB, UB, OPTIONS)
FUN Function handle Objective function to be minimized.
⊕ X0 1 × M Double
default: (LB+UB)/2
Starting point of the optimization
algorithm.
⊕ SIGMA0 Scalar or 1 × M Double
default: (UB-LB)/6
Initial global step size.
⊕ LB Scalar or 1 × M Double
default: -Inf
Lower bounds of the design space.
Continued on next page
- 22 - uq ceo UQLAB-V2.1-201
UQLIB – Optimization
Table 1–continued from previous page
⊕ UB Scalar or 1 × M Double
default: Inf
Upper bounds of the design space.
OPTIONS Structure
default: Table 2
Options of the algorithm.
Table 2: uq_ceo(..., OPTIONS)
.Display String
default: 'final'
Level of output display.
'none' Display no output.
'iter' Display output at each iteration.
'final' Display only the final output.
.isVectorized Logical
default: true
Specifies whether the objective function
is vectorized.
true The objective function is vectorized.
false The objective function is not vectorized.
.MaxIter Integer
default: 100 · M
Maximum number of generations.
.nStallMax Integer
default: 50
Maximum number of stall generations.
.MaxFunEval Positive Integer
default: Inf
Maximum number of objective function
evaluations.
.TolFun Double
default: 10
−3
Tolerance on the objective function: the
algorithm stops if the range (i.e., the
absolute difference between the
maximum and minimum values) of the
best objective function values over
.nStallMax generations is less than or
equal to .TolFun.
.TolSigma Double
default: 10
−3
Tolerance on the input variable: the
algorithm stops when σ
[t]
x
/σ
[0]
x
is
smaller or equal to .TolSigma.
.FvalMin Double
default: -Inf
Minimum cost: the algorithm stops
when the objective function is smaller
or equal to FvalMin.
.nPop Integer
default: 100
Population size.
.quantElite Integer
default: 0.05
Proportion of nPop that will be used as
elite sample (ρ in Section 2).
Continued on next page
UQLAB-V2.1-201 uq ceo - 23 -
UQLIB User manual
Table 2–continued from previous page
.alpha Double
default: 0.4
Smoothing parameter α
CE
(Eq. (3)).
.beta Double
default: 0.4
Smoothing parameter β
CE
(Eqs. (3)
(4)).
.q Double
default: 10
Smoothing exponent q
CE
(Eq. (4)).
6 Output
Table 3: [XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_ceo(...)
XSTAR 1 × M Double Optimal solution.
FSTAR Double Objective function value at the optimal
solution.
EXITFLAG Scalar Flag indicating the termination
condition of the algorithm.
1 Maximum number of generations
reached.
2 Maximum number of stall generations
reached.
3 Maximum number of function
evaluations reached.
4 Range of FUN over generations is
smaller than threshold.
5 Global step size smaller than threshold.
6 Minimum objective function value
reached.
<0 No feasible solution was found.
OUTPUT Structure, see Table 4 Diverse information about the
optimization process.
Table 4: [...,OUTPUT] = uq_ceo(...)
.message String Exit message.
.iterations Integer Total number of generations.
Continued on next page
- 24 - uq ceo UQLAB-V2.1-201
UQLIB – Optimization
Table 4–continued from previous page
.funccount Integer Total number of objective function
evaluations.
.History Structure, see Table 5 History of the optimization process over
iterations.
.lastgeneration Structure, see Table 6 Parameters of the last generation.
Table 5: OUTPUT.History
.Xmean Matrix Double History of the mean of the Gaussian
distribution at each iteration.
.sigma Matrix Double History of the global step size at each
iteration.
.Xbest Matrix Double History of the best sampled point at
each iteration.
.fitbest Vector Double History of the best objective function
value at each iteration.
.fitmedian Vector Double History of the median of the objective
function values at each iteration.
Table 6: OUTPUT.lastgeneration
.Xmean 1 × M Double Mean of the final Gaussian distribution.
.Xbest 1 × M Double Best solution from the last generation.
.bestfitness Double Best objective function value from the
last generation.
7 Notes
• The default values for X0 and SIGMA0 (see Table 1) are heuristics.
• The last five values in Table 2 are the CEO-specific parameters. The default values
selected there are based on a typical use of CEO for optimizing the hyperparameters of
Support Vector Machines MODEL in UQLAB (Moustapha et al., 2021b,a).
UQLAB-V2.1-201 uq ceo - 25 -
uq cmaes – Covariance Matrix
Adaptation Evolution Strategy
(CMA-ES)
1 Objective
Solve the following unconstrained optimization problem:
x
∗
= arg min
x∈D
X
f (x) , (1)
where x ∈ D
X
⊆ R is an M-dimensional vector; D
X
=
Q
M
i=1
x
lb
i
, x
ub
i
represents the search
space, with the lower and upper bounds of the i-th input dimension x
lb
i
and x
ub
i
, respectively;
x
∗
is the optimal solution; and f is a scalar-valued objective function.
2 Algorithm
The Covariance Matrix Adaptation–Evolution Strategy (CMA-ES) is a derandomized stochas-
tic search algorithm introduced by Hansen and Ostermeier (2001). The basic idea of the
algorithm is to sample points in the search space and adapting the sampling mechanism so
as to iteratively move towards the optimal solution. In practice, a Gaussian distribution with
a given covariance matrix is considered for sampling. The covariance matrix of the distribu-
tion is adapted at each iteration, such that the directions that have improved the objective
function in the recent past iterations are more likely to be sampled again. The mean is up-
dated considering a subset of the current iteration best samples (a.k.a. elite samples) using a
recombination scheme with predefined and possibly uneven weights.
Many versions of the algorithms exist, mostly with different selection mechanisms. UQLIB
provides the so-called (µ, λ)-CMA-ES. In the (µ, λ)-CMA-ES strategy, the candidate solutions
of the next generation consist of λ points sampled from a Gaussian distribution considering
only the µ best points of the current generation (Hansen and Kern, 2004; Hansen, 2001).
The actual implementation is more complex; the user can refer to Hansen (2001) for details.
The important steps of the algorithm are as follows:
26
UQLIB – Optimization
1. Initialize the algorithm:
• Set the parameters of the initial Gaussian distribution: the mean µ
[0]
x
∈ R
M
and
the standard deviation σ
[0]
x
∈ R
M
>0
which correspond to the algorithm starting
point and the step size, respectively. The parameter σ
[0]
= max
σ
[0]
x
corresponds
to the global step size.
• Initialize the covariance matrix C
[0]
= I
M
, where I
M
is an M ×M identity matrix.
• Set the internal CMA-ES parameters: the recombination scheme (details are given
in Table 2) of the weights {w
i
, i = 1, . . . , µ} (see Eq. (3)); the initial evolution
paths p
s
and p
c
; and the coefficients c
σ
, d
σ
, c
c
, c
cov
, and µ
cov
.
• Set t = 1, where t is the counter for the algorithm iteration (or so-called generation
in CMA-ES optimization).
2. Sample λ points
x
(1)
, x
(i)
, . . . , x
(λ)
such that
x
(i)
= µ
[t−1]
x
+ σ
[t−1]
B
[t−1]
D
[t−1]
z
(i)
, (2)
where z
(i)
∼ N (0, I
M
); and B and D are obtained from the eigendecomposition of
the covariance matrix C
[t−1]
, i.e., C
[t−1]
=
B
[t−1]
T
D
[t−1]
2
B
[t−1]
.
3. Evaluate the objective function at the sample points f
x
(i)
, i = 1, . . . , λ.
4. Select µ sample points with the smallest value of the objective function. Those sample
points are denoted by
x
(1)
, . . . , x
(µ)
.
5. Update the mean of the Gaussian distribution:
µ
[t]
x
=
µ
X
i=1
w
i
x
(i)
, (3)
where w
i
are weights whose values depend on the recombination scheme (see .recombination
in Table 2).
6. Update the global step size:
σ
[t]
= σ
[t−1]
exp
c
σ
d
σ
kp
[t]
s
k
√
M
1 −
1
4M
+
1
21M
2
− 1
!!
, (4)
where k·k denotes the Euclidean norm and
p
[t]
s
= (1 − c
s
)p
[t−1]
s
+
p
c
s
(2 − c
s
) µ
eff
B
[t−1]
µ
[t−1]
z
, (5)
where µ
eff
= 1/
P
µ
i=1
w
2
i
is the so-called variance effective selection mass; and µ
[t−1]
z
=
P
µ
i=1
w
i
z
(i)
.
UQLAB-V2.1-201 uq cmaes - 27 -
UQLIB User manual
7. Update the covariance matrix:
C
[t]
= (1 − c
cov
) C
[t−1]
+
c
cov
µ
cov
p
[t]
c
p
[t]
c
T
+ c
cov
1 −
1
µ
cov
µ
X
i=1
w
i
σ
[t−1]
2
x
(i)
− µ
[t−1]
x
x
(i)
− µ
[t−1]
x
T
.
(6)
where
p
[t]
c
= (1 − c
c
)p
[t−1]
c
+ h
σ
p
c
c
(2 − c
c
) µ
eff
σ
[t−1]
µ
[t]
x
− µ
[t−1]
x
, (7)
and h
σ
is defined as
h
σ
=
1, if
r
kp
[t]
s
k
1−(1−c
σ
)
2(g/λ)
√
M
1 −
1
4M
+
1
21M
2
< 1.4 +
2
M+1
0, otherwise
, (8)
where g is the current number of objective function evaluations.
8. If convergence is achieved, stop the algorithm; otherwise increase t ← t + 1 and go
back to Step 2. The following convergence criteria are considered:
• Number of generations: the algorithm stops if the number of generations (i.e.,
iterations) reaches a given threshold.
• Number of stall generations: the algorithm stops if the number of successive iter-
ations without sampling a point that improves the current best solution reaches a
given threshold.
• Number of function evaluations: the algorithm stops if the number of calls to the
objective function reaches a given threshold.
• Stagnation of the cost: the algorithm stops if the absolute difference between the
maximum and minimum of the objective function values over a given number of
iterations (i.e., its range) is below a given threshold.
• Stagnation of solution: the algorithm stops if the possible change in the solution
becomes extremely small, i.e., it falls below a given threshold.
The algorithm stops when any one of these criteria is reached.
3 Syntax
XSTAR = uq_cmaes(FUN, X0, SIGMA0)
XSTAR = uq_cmaes(FUN, X0, SIGMA0, LB, UB)
XSTAR = uq_cmaes(FUN, X0, SIGMA0, LB, UB, OPTIONS)
[XSTAR,FSTAR] = uq_cmaes(...)
[XSTAR,FSTAR,EXITFLAG] = uq_cmaes(...)
[XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_cmaes(...)
XSTAR = uq_cmaes(FUN, X0, SIGMA0) finds a local minimizer of the function FUN with
X0 as the starting point and SIGMA0 as the initial coordinate-wise standard deviation.
- 28 - uq cmaes UQLAB-V2.1-201
UQLIB – Optimization
XSTAR = uq_cmaes(FUN, X0, SIGMA0, LB, UB) defines a set of lower and upper bounds
such that LB(i) <= XSTAR(i) <= UB(i). If LB and UB are finite and X0 = [] and/or
SIGMA0 = [], the center of the search space (LB(i)+UB(i))/2 and 1/6 of the search
space width, i.e., (UB(i)-LB(i))/6 are used as X0(i) and SIGMA0(i), respectively.
XSTAR = uq_cmaes(FUN, X0, SIGMA0, LB, UB, OPTIONS) minimizes with the default
optimization options replaced by the values in the OPTIONS structure (see Table 2).
[XSTAR,FSTAR] = uq_cmaes(...) additionally returns the value of the objective function
FSTAR at the solution XSTAR.
[XSTAR,FSTAR,EXITFLAG] = uq_cmaes(...) additionally returns an exit flag that indi-
cates the termination condition of the algorithm (see Table 4).
[XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_cmaes(...) additionally returns a structure with
additional information about the optimization process (see Table 4).
4 Examples
4.1 Minimize Rosenbrock’s function
Consider the minimization problem of the Rosenbrock’s function:
x
∗
= arg min
x∈[−10,10]
2
100(x
2
− x
2
1
)
2
+ (1 − x
1
)
2
. (9)
The minimum of this function is located at x
∗
= [1, 1] with the minimum value f
∗
= 0.
The following code solves the optimization problem using uq_cmaes by assuming default
values for the starting point X0 and initial global step size SIGMA0 (see Table 5):
rng(100, 'twister') % Fo r r e p r o d u c i b l e r e s u l t s
fun = @(X) 100 .
*
(X(:,2) - X(:,1).ˆ2).ˆ2 + (1 - X(:,1)).ˆ2;
lb = [-10 -10];
ub = [10 10];
xstar = uq_cmaes(fun, [], [], lb, ub)
The code produces:
Possible change in X is below options.TolX
obj. value = 1.03161e-22
xstar =
1.0000 1.0000
UQLAB-V2.1-201 uq cmaes - 29 -
UQLIB User manual
4.2 Specify the starting point and initial global step size
By default, the starting point of the uq_cmaes is X0(i) = (LB(i)+UB(i))/2 while the initial
global step size is SIGMA0(i) = (LB(i)+UB(i))/6. The following code specifies user-given
values for these two parameters:
rng(100, 'twister') % Fo r r e p r o d u c i b l e r e s u l t s
fun = @(X) 100 .
*
(X(:,2) - X(:,1).ˆ2).ˆ2 + (1 - X(:;1)).ˆ2;
lb = [-10 -10];
ub = [10 10];
x0 = [-5 5];
sigma0 = [2 2];
xstar = uq_cmaes(fun, x0, sigma0, lb, ub)
The code produces:
Possible change in X is below options.TolX
obj. value = 5.5885e-24
xstar =
1.0000 1.0000
5 Input
Table 1: uq_cmaes(FUN, X0, SIGMA0, LB, UB, OPTIONS)
FUN Function handle Objective function to be minimized.
⊕ X0 1 × M Double
default: (LB+UB)/2
Starting point of the algorithm.
⊕ SIGMA0 1 × M Double
default: (UB-LB)/6
Initial global step size.
⊕ LB Scalar or 1 × M Double
default: -Inf
Lower bounds of the design space.
⊕ UB Scalar or 1 × M Double
default: Inf
Upper bounds of the design space.
OPTIONS Structure, see Table 2 Options of the algorithm.
Table 2: uq_cmaes(..., OPTIONS)
.lambda Positive Integer
default: 4 + b3 log M c
Population size (λ).
.mu Positive Integer
default: bλ/2c
Parent numbers (µ).
Continued on next page
- 30 - uq cmaes UQLAB-V2.1-201
UQLIB – Optimization
Table 2–continued from previous page
.recombination String
default:
'superlinear'
Computation of the weights w
i
used for
the recombination in Eq. (3). The
weights are normalized before
recombination: w
i
= bw
i
/
P
µ
i
bw
i
.
'equal' Assigns same weight to all parents
regardless of their rank: bw
i
= 1/µ.
'linear' Assigns weights that varies linearly with
respect to the parents rank:
bw
i
= µ + 1/2 − i.
'superlinear' Assigns weights that vary superlinearly
with respect to the parents rank:
bw
i
= log(µ + 1/2) − log(i).
.boundsHandling String
default: 'resampling'
Strategy how to handle out of bounds
samples.
'resampling' Resamples out-of-bound sample points.
'penalization' Projects out-of-bounds sample points
and penalize the corresponding
objective (fitness) function values.
.Display String
default: 'final'
Level of display.
'none' Displays no output.
'iter' Displays output at each iteration.
'final' Displays only the final output.
.MaxIter Positive integer
default:
b10
3
· (M + 5)
2
/
√
λc
Maximum number of generations.
.nStallMax Positive integer
default:
max (70, 10 + d30 · M/λe)
Maximum number of stall generations.
.TolFun Double
default: 10
−12
Tolerance on the objective function: the
algorithm stops if the range (i.e., the
absolute difference between the
maximum and minimum values) of the
best objective function values over
.nStallMax generations is less than or
equal to .TolFun.
.TolX Double
default: 10
−11
· max(σ
[0]
x
)
Tolerance on the input x: the algorithm
stops when the global step size is too
small to allow sampling far enough
from the current minimum.
Continued on next page
UQLAB-V2.1-201 uq cmaes - 31 -
UQLIB User manual
Table 2–continued from previous page
.MaxFunEval Positive Integer
default: Inf
Maximum number of function
evaluations.
.isVectorized Logical
default: true
Specify if the objective function is
vectorized.
.keepCDiagonal Integer
default: 0
Specify if the covariance matrix should
be kept diagonal.
≤ 0 Covariance matrix is never kept
diagonal.
1 Covariance matrix is always kept
diagonal.
> 1 Covariance matrix is kept diagonal for
the first .keepCDiagonal iterations.
.isActiveCMA Logical
default: true
Specify if the covariance matrix should
be actively adapted (updated), when
the current path leads repeatedly to
unsuccessful sample points (i.e., sample
points that do not improve the current
best solution).
.Strategy Structure
default: Table 3
Internal parameters of the CMA-ES
algorithm. They are already optimized.
It is strongly advised not to modify
them.
Table 3: OPTIONS.Strategy
.cs Double
default:
µ
eff
M+µ
eff
+5
c
σ
in Section 2. See Hansen (2001) for
details.
.ds Double
default:
d
σ
in Section 2. See Hansen (2001) for
details.
1+2·max
0,
q
µ
eff
−1
M+1
+c
σ
.cc Double
default:
4+µ
eff
/M
M+4+2·µ
eff
/M
c
c
in Section 2. See Hansen (2001) for
details.
.c1 Double
default:
2
(M+1.3)
2
+µ
eff
c
cov
in Section 2. See Hansen (2001) for
details.
.cmu Double
default:
µ
cov
in Section 2. See Hansen (2001)
for details.
min
1−c
cov
,2·
µ
eff
−2+1/µ
eff
(M+2)
2
+µ
eff
where µ
eff
= 1/
P
µ
i=1
w
2
i
is the variance effective selection mass.
- 32 - uq cmaes UQLAB-V2.1-201
UQLIB – Optimization
Note: These five parameters of the CMA-ES algorithm have been fine-tuned according
to Hansen and Ostermeier (2001). It is not advised to modify their default values.
6 Output
Table 4: [XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_cmaes(...)
XSTAR 1 × M Double Optimal solution.
FSTAR Double Value of the objective function at the
optimal solution.
EXITFLAG Integer Flag indicating the termination
condition of the algorithm.
1 Maximum number of generations is
reached.
2 Maximum number of stall generations is
reached.
3 Maximum number of function
evaluations is reached.
4 Range of FUN over generations is
smaller than threshold.
5 Current global step size is smaller than
threshold.
<0 No feasible solution was found.
OUTPUT Structure, see Table 5 Diverse information about the
optimization process.
Table 5: [...,OUTPUT] = uq_cmaes(...)
.message String Exit message.
.iterations Integer Total number of generations.
.funccount Integer Total number of objective function
evaluations.
.History Structure, see Table 6 History of the optimization process over
iterations.
.lastgeneration Structure, see Table 7 Parameters of the last generation.
UQLAB-V2.1-201 uq cmaes - 33 -
UQLIB User manual
Table 6: OUTPUT.History
.Xmean Matrix Double History of the mean of the Gaussian
distribution at each iteration.
.sigma Vector Double History of the global step size at each
iteration.
.Xbest Vector Double History of the best sampled point at
each iteration.
.fitbest Vector Double History of the best objective function
value at each iteration.
.fitmedian Double vector History of the median of the objective
function values at each iteration.
Table 7: OUTPUT.lastgeneration
.Xmean 1 × M Double Mean of the final Gaussian distribution
.Xbest 1 × M Double Best solution from the last generation
.bestfitness Vector Double Best objective function from the last
generation.
7 Notes
• uq_cmaes is a simple implementation of CMA-ES and it is suited for solving optimiza-
tion problems in UQLab. The code has not been optimized yet. For a better perfor-
mance, refer to the implementation in the CMA-ES website (last accessed: 06/11/2018).
• Some numerical considerations (e.g., conditioning of the covariance matrix) in uq_cmaes
are directly taken from the original MATLAB CMA-ES implementation by Hansen (2001)
available in the CMAE-ES website (last accessed: 06/11/2018).
• The default values for X0 and SIGMA0 (see Table 1) are heuristics.
• The parameters p
s
and p
c
are both initialized to 0
M
, an M-dimensional vector of zero.
- 34 - uq cmaes UQLAB-V2.1-201
uq 1p1cmaes – (1+1)-CMAES
1 Objective
Solve the following unconstrained optimization problem:
x
∗
= arg min
x∈D
X
f (x) , (1)
where x ∈ D
X
⊆ R is an M-dimensional vector; D
X
=
Q
M
i=1
x
lb
i
, x
ub
i
represents the search
space, with the lower and upper bounds of the i-th input dimension x
lb
i
and x
ub
i
, respectively;
x
∗
is the optimal solution; and f is a scalar-valued objective function.
2 Algorithm
The (1+1)-CMA-ES algorithm is a variant of the covariance matrix adaptation–evolution
strategy (CMA-ES) developed by Igel et al. (2006) and Suttorp et al. (2009) (see uq_cmaes).
In this variant, one parent generates exactly one offspring and the covariance matrix is
adapted so as to favor sampling in the directions that improve the objective function.
uq_1p1cmaes follows the algorithm developed in Arnold and Hansen (2010) where an active
covariance matrix adaptation is proposed, i.e., the covariance matrix is updated considering
both successful and unsuccessful trial steps. Moreover, the algorithm directly resorts to an
incremental update of the Cholesky decomposition of the covariance matrix. Both aspects
make the algorithm efficient.
Details of the implementation are shown in the following (Arnold and Hansen, 2010):
1. Initialize the algorithm:
• Set the the starting point x
best
= x
[0]
and the global step size σ.
• Initialize the state parameters of the algorithm: the search path s ∈ R
M
; the
success probability estimate P
succ
∈ [0, 1]; the Cholesky factor A from the decom-
position of the covariance matrix C (i.e., C = AA
T
), and its inverse A
inv
(both
are initialized to be the M × M identity matrix).
• Set the internal CMA-ES parameters: d
p
, c
p
, c
c
, c
+
cov
, P
target
, and P
thres
.
• Set t = 1, where t is the counter for the algorithm iteration (or so-called generation
in CMA-ES-based optimization).
35
UQLIB User manual
2. Generate an offspring candidate:
x
[t]
= x
best
+ σAz, (2)
where z ∼ N
M
(0, I
M
) and I
M
is an M × M identity matrix.
3. Evaluate the objective function at the offspring candidate f
x
[t]
:
• If f
x
[t]
≤ f (x
best
):
(a) Update the current best solution x
best
← x
[t]
.
(b) Update the success probability estimate:
P
succ
← (1 − c
p
)P
succ
+ c
p
. (3)
(c) Update the search path:
– If P
succ
< P
thres
:
s ← (1 − c
c
)s +
p
c
c
(2 − c
c
)z and α = 1 − c
+
cov
. (4)
– Otherwise:
s ← (1 − c
c
)s and α =
1 − c
+
cov
+ c
+
cov
· c
c
(2 − c
c
) . (5)
(d) Set w = A
inv
s and update the Cholesky factor and its inverse:
A ←
√
αA +
√
α
kwk
2
s
1 +
c
+
cov
α
kwk
2
− 1
sw
T
,
A
inv
←
1
√
α
A
inv
−
1
√
αkwk
2
1 −
1
q
1 +
c
+
cov
α
kwk
2
w
w
T
A
inv
,
(6)
where k·k denotes the Euclidean norm.
(e) Go to Step 4
• Otherwise:
(a) Update the success probability estimate:
P
succ
← (1 − c
p
)P
succ
. (7)
(b) If x
[t]
is worse than its 5-th-order ancestor, i.e., f
x
[t]
≥ f
x
[t−4]
, update A
and A
inv
according to Eq. (6) while replacing c
+
cov
with c
−
cov
where
c
−
cov
= min
0.4
M
1.6
+ 1
,
1
2kzk
2
− 1
. (8)
(c) Go to Step 4
- 36 - uq 1p1cmaes UQLAB-V2.1-201
UQLIB – Optimization
4. Update the global step size:
σ ← σ exp
1
d
p
P
succ
− P
target
1 − P
target
. (9)
5. If convergence is achieved, stop the algorithm; otherwise increase t ← t + 1 and go
back to Step 2. The following convergence criteria are considered:
• Maximum number of generations: the algorithm stops if the number of genera-
tions (i.e., iterations) reaches a given threshold.
• Number of stall generations: the algorithm stops if the number of successive iter-
ations without sampling a point that improves the current best solution reaches a
given threshold.
• Stagnation of the cost: the algorithm stops if the absolute difference between the
maximum and minimum of the objective function values over a given number of
iterations (i.e., its range) is below a given threshold.
• Stagnation of the solution: the algorithm stops if the possible change in the solu-
tion becomes extremely small, i.e., the current normal distribution can only sample
points that are extremely close to its mean.
• Number of function evaluations: the algorithm stops if the number of calls to the
objective function reaches a given threshold.
The algorithm stops when any one of these criteria is reached.
3 Syntax
XSTAR = uq_1p1cmaes(FUN, X0, SIGMA0)
XSTAR = uq_1p1cmaes(FUN, X0, SIGMA0, LB, UB)
XSTAR = uq_1p1cmaes(FUN, X0, SIGMA0, LB, UB, OPTIONS)
[XSTAR,FSTAR] = uq_1p1cmaes(...)
[XSTAR,FSTAR,EXITFLAG] = uq_1p1cmaes(...)
[XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_1p1cmaes(...)
XSTAR = uq_1p1cmaes(FUN, X0, SIGMA0) finds a local minimizer of the function FUN
with X0 as the starting point and SIGMA0 as the initial global step size.
XSTAR = uq_1p1cmaes(FUN, X0, SIGMA0, LB, UB) defines a set of lower and upper
bounds such that LB(i) <= XSTAR(i) <= UB(i). If LB(i) and UB(i) are finite and
X0 = [] and/or SIGMA0 = [], the center of the search space (LB(i)+UB(i))/2 and
1/6 of the search space width, i.e., (UB(i)-LB(i))/6 are used as X0(i) and SIGMA0(i),
respectively.
XSTAR = uq_1p1cmaes(FUN, X0, SIGMA0, LB, UB, OPTIONS) minimizes with the de-
fault optimization options replaced by the values in the OPTIONS structure (see Ta-
ble 2).
[XSTAR,FSTAR] = uq_1p1cmaes(...) additionally returns the value of the objective func-
UQLAB-V2.1-201 uq 1p1cmaes - 37 -
UQLIB User manual
tion at the solution XSTAR.
[XSTAR,FSTAR,EXITFLAG] = uq_1p1cmaes(...) additionally returns an exit flag that in-
dicates the termination condition of the algorithm (see Table 4).
[XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_1p1cmaes(...) additionally returns a structure
with additional information about the optimization process (see Table 4).
4 Examples
4.1 Minimize Rosenbrock’s function
Consider the minimization problem of the Rosenbrock’s function:
x
∗
= arg min
x∈[−10,10]
2
100(x
2
− x
2
1
)
2
+ (1 − x
1
)
2
. (10)
The minimum of this function is located at x
∗
= (1, 1) with the minimum value f
∗
= 0.
The following code solves the optimization problem using uq_1p1cmaes by assuming default
values for the starting point X0 and initial global step size SIGMA0 (see Table 5):
rng(123,'twister') % For r e p r o d u c i b l e r e s u l t s
fun = @(X) 100 .
*
(X(:,2) - X(:,1).ˆ2).ˆ2 + (1 - X(:,1)).ˆ2;
lb = [-10 -10];
ub = [10 10];
xstar = uq_1p1cmaes(fun, [], [], lb, ub)
The code produces:
The relative change of F was below options.TolFun
obj. value = 1.35004e-17
xstar =
1.0000 1.0000
4.2 Specify the starting point and initial global step size
By default, the starting point of the uq_1p1cmaes is X0(i) = (LB(i)+UB(i))/2 and the
initial global step size is SIGMA0(i) = (LB(i)+UB(i))/6. The following code specifies user-
given values for these two parameters:
rng(123,'twister') % For r e p r o d u c i b l e r e s u l t s
fun = @(X) 100 .
*
(X(:,2) - X(:,1).ˆ2).ˆ2 + (1 - X(:,1)).ˆ2;
lb = [-10 -10];
ub = [10 10];
x0 = [-5 5];
sigma0 = [2 2];
xstar = uq_1p1cmaes(fun, x0, sigma0, lb, ub);
- 38 - uq 1p1cmaes UQLAB-V2.1-201
UQLIB – Optimization
The code produces:
The relative change of F was below options.TolFun
obj. value = 7.47186e-13
xstar =
1.0000 1.0000
5 Input
Table 1: uq_1p1cmaes(FUN, X0, SIGMA0, LB, UB, OPTIONS)
FUN Function handle Objective function to be minimized.
⊕ X0 1 × M Double
default: (LB+UB)/2
Starting point of the optimization
algorithm.
⊕ SIGMA0 Scalar or 1 × M Double
default: (UB-LB)/6
Initial global step size.
⊕ LB Scalar or 1 × M Double
default: -Inf
Lower bounds of the search space.
⊕ UB Scalar or 1 × M Double
default: Inf
Upper bounds of the search space.
OPTIONS Structure
default: Table 2
Options of the algorithm.
Table 2: uq_1p1cmaes(..., OPTIONS)
.Display String
default: 'final'
Level of output display.
'none' Displays no output.
'iter' Displays output at each iteration.
'final' Displays only the final output.
.MaxIter Double
default: 1000 (M + 5)
2
Maximum number of generations
(iterations).
.nStallMax Positive integer
default: 50
Maximum number of stall generations.
.MaxFunEval Positive Integer
default: Inf
Maximum number of function
evaluations.
Continued on next page
UQLAB-V2.1-201 uq 1p1cmaes - 39 -
UQLIB User manual
Table 2–continued from previous page
.TolFun Double
default: 10
−6
Tolerance on the objective function: the
algorithm stops if the range (i.e., the
absolute difference between the
maximum and minimum values) of the
best objective function values over
.nStallMax generations is less than or
equal to .TolFun.
.TolSigma Double
default: 10
−11
· σ
[0]
x
Tolerance on the input x: the algorithm
stops when the current global step size
σ is lower than .TolSigma.
.isActiveCMA Logical
default: true
Specify if the covariance matrix should
be actively adapted (updated), when
the current path leads repeatedly to
unsuccessful sample points (i.e., sample
points that do not improve the current
best solution).
.Strategy Structure
default: Table 3
Internal parameters of the
(1+1)-CMA-ES algorithm. They are
already optimized. It is strongly advised
not to modify them.
Table 3: OPTIONS.Strategy
.dp Double
default: 1 + M/2
d
p
in Eq. (9), see Suttorp et al. (2009)
for details.
.Ptarget Double
default: 2/11
P
target
in Eq. (9), see Suttorp et al.
(2009) for details.
.cp Double
default: 1/12
c
p
in Eqs. (3) and (7), see Suttorp et al.
(2009) for details.
.cc Double
default: 2/(M + 2)
c
c
in Eqs. (4) and (5), see Suttorp et al.
(2009) for details.
.ccov Double
default: 2/(M
2
+ 6)
c
+
cov
in Eqs. (4 – 6), see Suttorp et al.
(2009) for details.
.Pthres Double
default: 0.44
P
thres
in Eq. (4), see Suttorp et al.
(2009) for details.
Note: These six parameters have been fine-tuned for the (1+1)-CMA-ES algorithm
(Suttorp et al., 2009). It is not advised to modify their default values.
- 40 - uq 1p1cmaes UQLAB-V2.1-201
UQLIB – Optimization
6 Output
Table 4: [XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_1p1cmaes(...)
XSTAR 1 × M Double Optimal solution.
FSTAR Double Objective function value at the optimal
solution.
EXITFLAG Scalar Flag indicating the termination
condition of the algorithm
1 Maximum number of generations
reached.
2 Maximum number of stall generations
reached.
3 Maximum number of function
evaluations reached.
4 Range of FUN over generations is
smaller than threshold.
5 Global step size sigma smaller than
threshold.
<0 No feasible solution was found.
OUTPUT Table 5 Diverse information about the
optimization process.
Table 5: [...,OUTPUT] = uq_1p1cmaes(...)
.message String Exit message.
.iterations Integer Total number of iterations.
.funccount Integer Total number of objective function
evaluations.
.History Structure, see Table 6 History of the optimization process over
iterations.
UQLAB-V2.1-201 uq 1p1cmaes - 41 -
UQLIB User manual
Table 6: OUTPUT.History
.x Matrix Double History of the sampled points at each
iteration.
.fval Vector Double History of the sampled objective
function values at each iteration.
.sigma Matrix Double History of the global step size at each
iteration.
7 Notes
• In the (1+1)-CMA-ES algorithm, the covariance matrix C itself is never explicitly com-
puted nor required.
• In uq_1p1cmaes, the initial values for the Cholesky factor A and its inverse A
inv
are
the identity matrix, which correspond to the unit-variance diagonal covariance matrix.
• The default values for X0 and SIGMA0 (see Table 1) are heuristics.
- 42 - uq 1p1cmaes UQLAB-V2.1-201
uq c1p1cmaes – Constrained
(1+1)-CMAES
1 Objective
Solve the following constrained optimization problem:
x
∗
= arg min
x∈D
X
f (x) subject to: g
k
(x) ≤ 0, k = 1, . . . , K, (1)
where x ∈ D
X
⊆ R is an M-dimensional vector; D
X
=
Q
M
i=1
x
lb
i
, x
ub
i
represents the search
space, with the lower and upper bounds of the i-th input dimension x
lb
i
and x
ub
i
, respectively;
x
∗
is the optimal solution; f is a scalar-valued objective function; and g
k
are K > 0 scalar-
valued constraint functions that need to be fulfilled.
2 Algorithm
The constrained (1+1)-CMA-ES algorithm is a variant of the (1+1)-CMA-ES (Arnold and
Hansen, 2010) algorithm developed by Arnold and Hansen (2012) where a constraint han-
dling scheme is added (see uq_1p1cmaes). (1+1)-CMA-ES itself is a variant of the covariance
matrix adaptation–evolution scheme (CMA-ES), where one parent generates exactly one off-
spring and the covariance matrix is adapted to favor sampling in the directions that improve
the objective function (see uq_cmaes). In the constrained version, the covariance matrix is
also updated when unfeasible points are sampled to decrease the probability of sampling
again in such directions.
uq_c1p1cmaes implementation follows the algorithm developed in Arnold and Hansen (2012)
where an active covariance matrix adaptation is proposed, i.e., the covariance matrix is up-
dated considering both successful and unsuccessful trial steps. Moreover, the algorithm di-
rectly resorts to an incremental update of the Cholesky decomposition of the covariance
matrix. Both aspects make the algorithm efficient.
Details of the implementation are shown in the following (Arnold and Hansen, 2012):
1. Initialize the algorithm:
• Set the starting point x
best
= x
[0]
and the global step size σ.
43
UQLIB User manual
• Initialize the state parameters of the algorithm: the search path s ∈ R
M
; the
success probability estimate P
succ
∈ [0, 1]; the Cholesky factor A from the decom-
position of the covariance matrix C (i.e., C = AA
T
), and its inverse A
inv
(both
are initialized to be the M ×M identity matrix); and a set of exponentially fading
v
k
, k = 1, . . . , K, initialized to be zeros.
• Set the internal CMA-ES parameters: c, d
p
, c
p
, c
c
, P
target
, c
+
cov
, and β.
• Set t = 1, where t is the counter for the algorithm iteration (or so-called generation
in CMA-ES-based optimization).
2. Generate an offspring candidate:
x
[t]
= x
best
+ σAz, (2)
where z ∼ N
M
(0, I
M
) and I
M
is an M × M identity matrix.
3. Evaluate the constraints at the offspring candidate g
k
x
[t]
, k = 1, . . . , K.
4. For all k = 1, . . . , K such that g
k
x
[t]
> 0, update v
k
:
v
k
← (1 − c) v
k
+ c
c
Az. (3)
5. If x
[t]
is not feasible (i.e.,
P
K
k=1
I
(
g
k
(
x
[t]
)
>0
)
≥ 1, where I
(◦)
denotes the indicator func-
tion defined in Eq. (5)):
(a) Update the Cholesky factor A:
A ← A −
β
P
K
k=1
I
(
g
k
(
x
[t]
)
>0
)
K
X
k=1
I
(
g
k
(
x
[t]
)
>0
)
v
k
w
T
k
w
T
k
w
k
(4)
where w
k
= A
−1
v
k
and I
(◦)
denotes the indicator function defined by:
I
(
g
k
(
x
[t]
>0
))
=
1 if g
k
x
[t]
> 0,
0 otherwise.
(5)
(b) Go back to Step 2.
6. Evaluate the objective function at the offspring candidate f
x
[t]
.
7. Update the success probability estimate:
P
succ
← (1 − c
p
)P
succ
+ c
p
I
(
f
(
x
[t]
)
≤f(x
best
)
)
. (6)
where I
(◦)
denotes the indicator function defined in Eq. (5).
- 44 - uq
c1p1cmaes UQLAB-V2.1-201
UQLIB – Optimization
8. Update the global step size:
σ ← σ exp
1
d
p
P
succ
− P
target
1 − P
target
. (7)
9. If f
x
[t]
≤ f (x
best
), go to Step 10; otherwise go to Step 14.
10. Set x
best
← x
[t]
.
11. Update the search path:
s ← (1 − c)s +
p
c(2 − c)Az. (8)
12. Set w = A
−1
s and update the Cholesky factor and its inverse:
A ← aA + bsw
T
,
A
inv
←
1
a
A
inv
−
b
a
2
+ abkwk
2
w
w
T
A
inv
,
(9)
where
a =
p
1 − c
+
cov
b =
p
1 − c
+
cov
kwk
2
s
1 +
c
+
cov
1 − c
+
cov
kwk
2
− 1
.
(10)
where k·k denotes the Euclidean norm.
13. Go to Step 15.
14. If x
[t]
is worse than its 5-th-order ancestor, i.e., f
x
[t]
≥ f
x
[t−4]
, set w = A
−1
s and
update A according to Eq. (9) with the coefficients a and b computed as follows:
a =
p
1 − c
−
cov
,
b =
p
1 − c
−
cov
kzk
2
s
1 +
c
−
cov
1 − c
−
cov
kzk
2
− 1
,
(11)
where
c
−
cov
= min
0.4
M
1.6
+ 1
,
1
2kzk
2
− 1
. (12)
15. If convergence is achieved, stop the algorithm; otherwise increase t ← t + 1 and go
back to Step 2. The following convergence criteria are considered:
• Maximum number of generations: the algorithm stops if the number of genera-
tions (i.e., iterations) reaches a given threshold.
• Number of stall generations: the algorithm stops if the number of successive iter-
ations without sampling a point that improves the current best solution reaches a
given threshold.
UQLAB-V2.1-201 uq c1p1cmaes - 45 -
UQLIB User manual
• Stagnation of the cost: the algorithm stops if the absolute difference between the
maximum and minimum of the objective function values over a given number of
iterations (i.e., its range) is below a given threshold.
• Stagnation of the solution: the algorithm stops if the possible change in the solu-
tion becomes extremely small, i.e., the current normal distribution can only sample
points that are extremely close to its mean.
• Number of function evaluations: the algorithm stops if the number of calls to the
objective function reaches a given threshold.
The algorithm stops when any one of these criteria is reached.
3 Syntax
XSTAR = uq_c1p1cmaes(FUN, X0, SIGMA0)
XSTAR = uq_c1p1cmaes(FUN, X0, SIGMA0, LB, UB)
XSTAR = uq_c1p1cmaes(FUN, X0, SIGMA0, LB, UB, NONLCON)
XSTAR = uq_c1p1cmaes(FUN, X0, SIGMA0, LB, UB, NONLCON, OPTIONS)
[XSTAR,FSTAR] = uq_c1p1cmaes(...)
[XSTAR,FSTAR,EXITFLAG] = uq_c1p1cmaes(...)
[XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_c1p1cmaes(...)
XSTAR = uq_c1p1cmaes(FUN, X0, SIGMA0) finds a local minimizer of the function FUN
with X0 as starting point and SIGMA0 as the initial global step size.
XSTAR = uq_c1p1cmaes(FUN, X0, SIGMA0, LB, UB) defines a set of lower and upper
bounds such that LB(i) <= XSTAR(i) <= UB(i). If LB and UB are finite and X0 = []
and/or SIGMA0 = [], the center of the search space, i.e., (LB(i)+UB(i))/2 and 1/6
of the search space width, i.e., (UB(i)-LB(i))/6 are used as X0(i) and SIGMA0(i),
respectively.
XSTAR = uq_c1p1cmaes(FUN, X0, SIGMA0, LB, UB, NONLCON) defines a set of non-linear
inequalities constraints and subjects the minimization to the constraints. If there are
no bound constraints, set LB = [] and UB = [].
XSTAR = uq_c1p1cmaes(FUN, X0, SIGMA0, LB, UB, NONLCON, OPTIONS) minimizes with
the default optimization options replaced by the values in the OPTIONS structure (see Ta-
ble 2).
[XSTAR,FSTAR] = uq_c1p1cmaes(...) additionally returns the value of the objective func-
tion at the solution XSTAR.
[XSTAR,FSTAR,EXITFLAG] = uq_c1p1cmaes(...) additionally returns an exit flag that
indicates the termination condition of the algorithm (see Table 4).
[XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_c1p1cmaes(...) additionally returns a struc-
ture with additional information about the optimization process (see Table 4).
- 46 - uq c1p1cmaes UQLAB-V2.1-201
UQLIB – Optimization
4 Examples
4.1 Minimize constrained Rosenbrock’s function
Consider a constrained minimization problem of the Rosenbrock’s function:
f(x) = 100
x
2
− x
2
1
2
+ (1 − x
1
)
2
,
subject to: g(x) = x
2
1
+ x
2
2
− 1.
(13)
The minimum of the function is located at x
∗
= (0.7864, 0.6177) with f (x
∗
) = 0.0457.
The following code solves the optimization problem using uq_c1p1cmaes by assuming de-
fault values for the starting point X0 and initial global step size SIGMA0 (see Table 5):
rng(42,'twister') % For r e p r o d u c i b l e r e s u l t s
fun = @(X) 100 .
*
(X(:,2) - X(:,1).ˆ2).ˆ2 + (1 - X(:,1)).ˆ2;
nonlcon = @(X) X(:,1).ˆ2 + X(:,2).ˆ2 - 1;
lb = [-10 -10];
ub = [10 10];
xstar = uq_c1p1cmaes(fun, [], [], lb, ub, nonlcon)
The code produces:
Value of sigma below options.TolSigma
obj. value = 0.0456748
xstar =
0.7864 0.6177
4.2 Specify the starting point and initial global step size
By default, the starting point of uq_c1p1cmaes is X0(i) = (LB(i)+UB(i))/2 and the initial
global step size is SIGMA0(i) = (UB(i)-LB(i))/6. The following code specifies user-given
values for these two parameters:
rng(100,'twister') % For r e p r o d u c i b l e r e s u l t s
fun = @(X) 100 .
*
(X(:,2) - X(:,1).ˆ2).ˆ2 + (1 - X(:;1)).ˆ2;
nonlcon = @(X) X(:,1).ˆ2 + X(:,2).ˆ2 - 1;
lb = [-10 -10];
ub = [10 10];
x0 = [0.5 0.5];
sigma0 = [2 2];
xstar = uq_c1p1cmaes(fun, x0, sigma0, lb, ub, nonlcon);
The code produces:
Value of sigma below options.TolSigma
obj. value = 0.0456748
UQLAB-V2.1-201 uq c1p1cmaes - 47 -
UQLIB User manual
xstar =
0.7864 0.6177
5 Input
Table 1: uq_c1p1cmaes(FUN, X0, SIGMA0, LB, UB, NONLCON, OPTIONS)
FUN Function handle Objective function to be minimized.
⊕ X0 1 × M Double
default: (LB+UB)/2
Starting point of the optimization
algorithm.
⊕ SIGMA0 Scalar or 1 × M Double
default: (UB-LB)/6
Initial global step size.
⊕ LB Scalar or 1 × M Double
default: -Inf
Lower bounds of the design space.
⊕ UB Scalar or 1 × M Double
default: Inf
Upper bounds of the design space.
NONLCON Function handle Nonlinear inequality constraints to
be satisfied.
OPTIONS Table 2 Options of the algorithm.
Table 2: uq_c1p1cmaes(..., OPTIONS)
.Display String
default: 'final'
Level of output display.
'none' Displays no output.
'iter' Displays output at each iteration.
'final' Displays only the final output.
.MaxIter Double
default: 1000 (M + 5)
2
Maximum number of iterations.
.MaxFunEval Positive Integer
default: Inf
Maximum number of function
evaluations.
.nStallMax Positive Integer
default: 10 + 30 · M
Maximum number of stall generations.
.TolFun Double
default: 10
−12
Tolerance on the objective function: the
algorithm stops if the range (i.e., the
absolute difference between the
maximum and minimum values) of the
best objective function values over
.nStallMax generations is less than or
equal to .TolFun.
Continued on next page
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UQLIB – Optimization
Table 2–continued from previous page
.TolSigma Double
default: 10
−11
· σ
[0]
x
Tolerance on the input x: the algorithm
stops if the current global step size
sigma is lower than .TolSigma.
.isactiveCMA Logical
default: true
Specify if the covariance matrix should
be actively adapted (updated), when
the current path leads repeatedly to
unsuccessful sample points (i.e., sample
points that do not improve the current
best solution).
.feasiblex0 Logical
default: true
If the starting point is not feasible,
search for a feasible point by random
sampling.
true Search for a feasible point before
proceeding to the algorithm.
false Proceed to the algorithm without
searching for a feasible point.
Strategy Structure
default: Table 3
Internal parameters of the
C(1+1)-CMA-ES algorithm. They are
already optimized. It is strongly advised
not to modify them.
Table 3: OPTIONS.Strategy
.dp Double
default: 1 + M/2
d
p
in Eq. (7), see Arnold and Hansen
(2012) for details.
.c Double
default: 2/(M + 2)
c in Eqs. (3) and (8), see Arnold and
Hansen (2012) for details.
.cp Double
default: 1/12
c
p
in Eq. (6), see Arnold and Hansen
(2012) for details.
.Ptarget Double
default: 2/11
P
target
in Eq. (7), see Arnold and
Hansen (2012) for details.
.ccovp Double
default: 2/(M
2
+ 6)
c
+
cov
in Eq. (11), see Arnold and Hansen
(2012) for details.
.cc Double
default: 1/(M + 2)
c
c
in Eq. (3), see Arnold and Hansen
(2012) for details.
.beta Double
default: 0.1/(M + 2)
β in Eq. (4), see Arnold and Hansen
(2012) for details.
Note: These seven parameters have been fine-tuned for the (1+1)-CMA-ES algorithm
(Arnold and Hansen, 2012; Suttorp et al., 2009). It is not advised to modify their
default values.
UQLAB-V2.1-201 uq c1p1cmaes - 49 -
UQLIB User manual
6 Output
Table 4: [XSTAR,FSTAR,EXITFLAG,OUTPUT] = uq_c1p1cmaes(...)
XSTAR 1 × M Double Optimal solution.
FSTAR Double Objective function value at the optimal
solution.
EXITFLAG Integer Flag indicating the termination
condition of the algorithm.
1 Maximum number of generations
reached.
2 Maximum number of stall generations
reached.
3 Maximum number of function
evaluations reached.
4 Range of FUN over generations is
smaller than threshold.
5 Global step size sigma smaller than
threshold.
<0 No feasible solution was found.
OUTPUT Structure, see Table 5 Diverse information about the
optimization process.
Table 5: [...,OUTPUT] = uq_c1p1cmaes(...)
.message String Exit message
.iterations Integer Total number of iterations.
.funccount Integer Total number of objective function
evaluations.
.constcount Integer Total number of constraint functions
evaluations.
.History Structure, see Table 6 History of the optimization process over
iterations.
Table 6: OUTPUT.History
.x Matrix Double History of the sampled points at each
iteration.
Continued on next page
- 50 - uq c1p1cmaes UQLAB-V2.1-201
UQLIB – Optimization
Table 6–continued from previous page
.fval Vector Double History of the corresponding objective
function values at each iteration.
.gval Matrix Double History of the corresponding constraint
function values at each iteration.
.sigma Matrix Double History of the global step size at each
iteration.
.status Vector Integer History of the state of the sampled point
at each iteration.
-1 Sampled point is not feasible.
0 Sampled point is feasible, but it does
not improve the current best solution.
1 Sampled point is feasible and it
improves the current best solution.
7 Notes
• In the constrained (1+1)-CMA-ES algorithm, the covariance matrix C itself is never
explicitly computed nor required.
• In uq_c1p1cmaes, the initial values for the Cholesky decomposition A and its inverse
A
inv
are the identity matrix, which correspond to the unit-variance diagonal covariance
matrix.
• The default values for X0 and SIGMA0 (see Table 1) are heuristics.
UQLAB-V2.1-201 uq c1p1cmaes - 51 -
uq eval Kernel – Compute kernel
matrix
1 Objective
Compute the kernel matrix given two input matrices X
1
and X
2
for a specified kernel func-
tion.
2 Algorithm
Some of the most popular kernels families are available in UQLAB. They are either classified
as stationary or non-stationary kernels.
2.1 Kernel matrix
Let X
1
and X
2
be matrices in R
N
1
×M
and R
N
2
×M
, respectively, whose row-wise elements are
row vectors x
(i)
1
and x
(i)
2
in D
X
⊆ R
M
. Let k be a kernel function, such that k : D
X
×D
X
7→ R.
Then the kernel matrix K is a N
1
× N
2
matrix defined by
K =
k
x
(1)
1
, x
(1)
2
k
x
(1)
1
, x
(2)
2
. . . k
x
(1)
1
, x
(N
2
)
2
k
x
(2)
1
, x
(1)
2
k
x
(2)
1
, x
(2)
2
. . . k
x
(2)
1
, x
(N
2
)
2
.
.
.
.
.
.
.
.
.
.
.
.
k
x
(N
1
)
1
, x
(1)
2
k
x
(N
1
)
1
, x
(2)
2
. . . k
x
(N
1
)
1
, x
(N
2
)
2
(1)
The kernel function k is any symmetric positive function that satisfies the Mercer’s condition
(Cherkassky and Mulier, 2007). If X
1
= X
2
, the resulting kernel matrix is the Gram matrix, a
positive-semidefinite matrix (Shawe-Taylor and Cristianini, 2004). In the context of Gaussian
process modeling, the kernel function and the resulting matrix correspond to the correlation
kernel function and matrix, respectively.
2.2 Available kernel function families
More comprehensive examples of a valid kernel function can be found in Vapnik (1995) or
Rasmussen and Williams (2006). The most popular ones are available in UQLAB and they
are listed below:
52
UQLIB – Optimization
• Stationary kernels: Stationary kernel functions depend only on the relative position
of its two inputs. All the one-dimensional families defined below are parametrized by
a kernel parameter θ, often referred to as the characteristic length scale parameter. The
kernel functions below are defined for a pair of one-dimensional input x, x
0
∈ R. Their
extension to multiple dimensions is given in Sections 2.3.1 and 2.3.2.
– Linear:
k
lin-s
x, x
0
; θ
= max
0, 1 −
|x − x
0
|
θ
. (2)
– Exponential:
k
exp
x, x
0
; θ
= exp
−
|x − x
0
|
θ
. (3)
– Gaussian (or squared exponential):
k
Gaussian
x, x
0
; θ
= exp
−
1
2
|x − x
0
|
θ
2
!
. (4)
– Mat
´
ern 3/2:
k
3/2
x, x
0
; θ
=
1 +
√
3 |x − x
0
|
θ
!
exp
−
√
3 |x − x
0
|
θ
!
. (5)
– Mat
´
ern 5/2:
k
5/2
x, x
0
; θ
=
1 +
√
5 |x − x
0
|
θ
+
5
3
|x − x
0
|
2
θ
2
!
exp
−
√
5 |x − x
0
|
θ
!
. (6)
– Custom user-defined stationary kernels: users can supply their own one-dimensional
stationary kernel using a function handle (see Section 2.3.3).
• Non-stationary kernels: Non-stationary kernel functions depend on the absolute val-
ues of the inputs, rather on their difference. In contrast with the stationary kernels
listed above, the following kernels are already defined in multi-dimension. That is,
they are defined for a pair of M-dimensional inputs x, x
0
∈ R
M
. Moreover, with
the exception of the non-stationary linear kernel function, non-stationary kernels are
parametrized by a vector of kernel parameters θ.
– Non-stationary linear:
k
lin
x, x
0
= x
T
x
0
. (7)
– Polynomial:
k
poly
x, x
0
; θ
=
x
T
x
0
+ d
p
, (8)
where θ = {d, p}, with d ≥ 0 and p ∈ N
∗
.
– Sigmoid:
k
sigmoid
x, x
0
; θ
= tanh
x
T
x
0
a
+ b
, (9)
where θ = {a, b}, with a > 0 and b ≤ 0.
UQLAB-V2.1-201 uq eval Kernel - 53 -
UQLIB User manual
2.3 Properties of stationary kernels
For stationary kernels which can be cast as a function of |x
1
−x
2
| (so-called radial basis form,
see Section 2.2), the following additional properties can be set by the user.
2.3.1 Kernel function types
When the input dimension M is greater than one, multi-dimensional stationary kernels can
be constructed from one-dimensional stationary kernel families using the following construc-
tions:
• Ellipsoidal kernel functions (Rasmussen and Williams, 2006), calculated as follows:
k
x, x
0
; θ
= k(h) , h =
v
u
u
t
M
X
i=1
x
i
− x
0
i
θ
i
2
. (10)
• Separable kernel functions (Sacks et al., 1989), calculated as follows:
k
x, x
0
; θ
=
M
Y
i=1
k
x
i
, x
0
i
, θ
i
. (11)
The function k(·) that appears on the right hand side of Eqs. (10) and (11) corresponds to
the available one-dimensional kernel function families described in Section 2.2. The types of
multi-dimensional construction above correspond to the anisotropic case, in which there is a
unique kernel parameter for each input dimension.
2.3.2 Isotropic kernels
A kernel function is called isotropic when a single kernel parameter is associated with all
the input dimensions. The isotropic version of the kernel function types introduced in the
previous section are given by:
• Isotropic ellipsoidal kernel functions:
k
x, x
0
; θ
= k
1
θ
v
u
u
t
M
X
i=1
(x
i
− x
0
i
)
2
, θ ∈ R. (12)
• Isotropic separable kernel functions:
k
x, x
0
; θ
=
M
Y
i=1
k(x
i
, x
0
i
; θ
, θ ∈ R. (13)
2.3.3 Custom user-defined kernels
A custom user-defined kernel function should return a valid stationary kernel and accept
three input arguments as illustrated in the function declaration below:
myK = my_eval_K(X1, X2, THETA)
- 54 - uq eval Kernel UQLAB-V2.1-201
UQLIB – Optimization
The inputs X1 and X2 are matrices with arbitrary number of rows and M number of columns
(M is also the input dimension). The input THETA corresponds to the length-scale pa-
rameter θ. A function handle referring to this function is then passed as an option to
uq_eval_Kernel.
Note: Custom user-defined kernels are only supported for stationary kernels.
2.4 Nugget
Regardless on how a kernel matrix K is calculated, it is often inverted in practical applica-
tions. This procedure is well known to suffer from numerical instabilities, especially when
the distances between the input points x
i
are small. To circumvent this limitation, one can
introduce a nugget ν, which is a set of values that are added to the main diagonal of K:
k
ii
= 1 + ν
i
. (14)
3 Syntax
K = uq_eval_Kernel(X1, X2, THETA, OPTIONS)
K = uq_eval_Kernel(X1, X2, THETA, OPTIONS) computes the kernel matrix K for two
inputs X1 and X2 given the kernel parameters THETA and additional options specified
in the structure OPTIONS (see Table 2).
4 Examples
4.1 Create a Gaussian correlation matrix
Compute the correlation matrix for the vector x =
0 0.25 0.50 0.75 1
T
, with a Gaus-
sian kernel with a correlation length of 0.25.
The following code creates the correlation matrix:
X = linspace(0,1,5)';
theta = 0.25;
Options.Family = 'Gaussian';
Options.Type = 'Separable';
Options.Isotropic = true;
Options.Nugget = 0;
R = uq_eval_Kernel(X, X, theta, OPTIONS)
The resulting matrix K is symmetric and has size 5 × 5:
R =
1.0000 0.6065 0.1353 0.0111 0.0003
0.6065 1.0000 0.6065 0.1353 0.0111
UQLAB-V2.1-201 uq eval Kernel - 55 -
UQLIB User manual
0.1353 0.6065 1.0000 0.6065 0.1353
0.0111 0.1353 0.6065 1.0000 0.6065
0.0003 0.0111 0.1353 0.6065 1.0000
Figure 1 compares three different correlation matrices computed by uq_eval_Kernel for
three different correlation lengths.
Figure 1: Plots of correlation matrices with three different correlation lengths θ: 0.1 (left),
0.25 (center), and 0.5 (right).
5 Input
Table 1: uq_eval_Kernel(X1, X2, THETA, OPTIONS)
X1 N
1
× M Double First input vector.
X2 N
2
× M Double Second input vector.
THETA Scalar or vector of Double Kernel function parameters, see
Section 2.2.
• For stationary and radial basis
type kernels: THETA should be
either a scalar or a vector of
size 1 × M.
• When THETA is scalar and the
Kernel is anisotropic the same
value is replicated in all
dimensions.
• For polynomial and sigmoid
kernels, θ should be a vector of
size 1 × 2.
OPTIONS Table 2 Options of the kernel function.
- 56 - uq eval Kernel UQLAB-V2.1-201
UQLIB – Optimization
Table 2: uq_eval_Kernel(..., OPTIONS)
.Family String or Function handle The kernel family, see Section 2.2.
'Linear' Linear kernel function, see Eq. (2).
'Exponential' Exponential kernel function, see Eq. (3).
'Gaussian' Gaussian kernel function, see Eq. (4).
'Matern-3_2' Mat
´
ern-3/2 kernel function, see Eq. (5).
'Matern-5_2' Mat
´
ern-5/2 kernel function, see Eq. (6).
'Linear-NS' Non-stationary linear kernel function,
see Eq. (7).
'Polynomial' Polynomial kernel function, see Eq. (8).
'Sigmoid' Sigmoid kernel function, see Eq. (9).
Function handle Custom user-defined stationary kernel
function, see Eq. (8).
.Type String Kernel function type. Only applies to
the stationary kernels, see Section 2.3.3.
'Ellipsoidal' Ellipsoidal kernel function.
'Separable' Separable kernel function.
.Isotropic Logical Determines whether the kernel function
is isotropic or anisotropic. Only applies
to the stationary kernel functions, see
Sections 2.2 and 2.3.2.
.Nugget Scalar or 1 × M Double Nugget value. Only applicable when X1
= X2 (i.e., a Gram matrix).
• If scalar, adds this quantity to the
diagonal elements of the kernel
matrix K.
• If vector, adds each element to
the corresponding diagonal
element of K.
6 Output
Table 3: K = uq_eval_Kernel(...)
K N
1
× N
2
Double Kernel matrix. A Gram matrix is
obtained when X1 = X2.
UQLAB-V2.1-201 uq eval Kernel - 57 -
uq subsample random – Random
subsampling
1 Objective
Given a sample set X =
x
(1)
, . . . , x
(N)
of size N , create a reduced set X
s
⊆ X by randomly
selecting N
s
≤ N sample points from X.
2 Algorithm
Random subsampling consists in creating a subset X
S
⊂ X by randomly selecting sample
points from X. That is, the subsampled experimental design X
s
contains a reduced number of
sample points X
s
=
x
(1)
, . . . , x
(N
s
)
, N
s
≤ N. This function may be used to create training
and validation sets for cross-validation in the context of machine learning algorithms.
3 Syntax
XS = uq_subsample_random(X,NS)
[XS,IDX] = uq_subsample_random(...)
XS = uq_subsample_random(X,NS) returns a subset XS (NS-by-M) of X (N-by-M), based on
random selection. NS has to be less than or equal to N.
[XS,IDX] = uq_subsample_random(...) additionally returns the indices of the selected
sample points, such that XS = X(IDX,:).
4 Input
Table 1: uq_subsample_random(X,NS)
X N × M Double Sample set.
NS Integer (N
s
≤ N) Size of the subsample.
58
UQLIB – Optimization
5 Output
Table 2: [XS,IDX] = uq_subsample_random(...)
XS N
s
× M Double Subsample.
IDX N
s
× M Integer Indices of the randomly selected
subsample points.
6 Example
This example creates a subsample the reduced Fisher’s Iris data set, a 100 × 2 data set. The
data set contains 100 observations and two variables, namely the petal width and length. The
reduced Fisher’s Iris data set is provided as part of the UQLAB distribution. The data set can
be loaded in variable X as follows:
load fisher_iris_reduced.mat
Out of the 100 sample points available in X, 10 are selected by random subsampling:
rng(100,'twister') % f o r r e p r o d u c i b l e r e s u l t s
Xs = uq_subsample_random(X,10);
Visualize the results by plotting the two different sets (see Figure 1):
uq_figure()
plot(X(:,1), X(:,2), '.')
hold on
plot(Xs(:,1), Xs(:,2), 'ro')
Figure 1: Random selection of 10 points from the reduced Fisher’ Iris data set by
uq_subsample_random.
UQLAB-V2.1-201 uq subsample random - 59 -
uq subsample kmeans – k-means
clustering-based subsampling
1 Objective
Given a sample set X =
x
(1)
, . . . , x
(N)
of size N, create a reduced set X
s
⊆ X by creating
N
s
≤ N clusters and, for each cluster, selecting the nearest point to the cluster centroid.
2 Algorithm
The k-means subsampling algorithm works as follows:
1. Identify N
s
clusters centroids in X using the k-means clustering algorithm with k = N
s
(Lloyd, 1982).
2. For each cluster centroid, the nearest sample point determined by the k-nearest-neighbor-
search algorithm with k = 1 is included in X
s
(Friedman et al., 1977).
3 Syntax
XS = uq_subsample_kmeans(X,NS)
XS = uq_subsample_kmeans(X, NS, NAME, VALUE)
[XS,IDX] = uq_subsample_kmeans(...)
XS = uq_subsample_kmeans(X,NS) returns a subset XS (NS-by-M) of X (N-by-M).
XS = uq_subsample_kmeans(X, NS, NAME, VALUE) allows for fine-tuning various pa-
rameters of the subsampling algorithm by specifying NAME and VALUE pairs of options.
The available options are summarized in Table 2.
[XS,IDX] = uq_subsample_kmeans(...) additionally returns the indices of the selected
sample points, such that XS = X(IDX,:).
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UQLIB – Optimization
4 Input
Table 1: uq_subsample_kmeans(X, NS, NAME, VALUE)
X N × M Double Sample set.
NS Integer (N
s
≤ N) Size of the subsample.
NAME, VALUE name-value pair, see
Table 2
Additional options.
Table 2: uq_subsample_kmeans(..., NAME, VALUE)
'Distance_kmeans' String
default:
'sqeuclidean'
Distance measure used in the k-means
clustering. List of the available options
can be found in the documentation of
the built-in MATLAB function kmeans
(see the option 'Distance').
'Distance_nn' String
default: 'euclidean'
Distance measure used in the
nearest-neighbor search for determining
the sample points closest to the k-means
centroids. List of available options can
be found in the documentation of the
built-in MATLAB function knnsearch
(see option 'Distance').
5 Output
Table 3: [XS,IDX] = uq_subsample_kmeans(...)
XS N
s
× M Double Subsample.
IDX N
s
× M Integer Indices of the randomly selected
subsample points.
6 Examples
This example subsamples the reduced Fisher’s Iris data set, a 100 × 2 data set. The data
set contains 100 observations and two variables, namely the petal width and length. The
reduced Fisher’s Iris data set is provided as part of the UQLAB distribution. The data set can
be loaded in variable X as follows:
load fisher_iris_reduced.mat
Out of the 100 sample points available in X, 10 are selected by k-means clustering using the
default options:
rng(100,'twister') % f o r r e p r o d u c i b l e r e s u l t s
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Xs = uq_subsample_random_kmeans(X,10);
Visualize the results by plotting the two different sets (see Figure 1):
uq_figure()
plot(X(:,1), X(:,2), '.')
hold on
plot(Xs(:,1), Xs(:,2), 'ro')
Figure 1: Selection of 10 points from the reduced Fisher’s Iris data set by
uq_subsample_kmeans.
7 Notes
• Strictly speaking, this method is more appropriately referred to as the k-medoid sam-
pling.
• This function utilizes MATLAB functions kmeans and knnsearch for the k-means clus-
tering and the nearest-neighbor search, respectively. Both functions are part of the
Statistics and Machine Learning Toolbox.
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uq figure – Create a figure
1 Objective
Create a figure window following the default formatting and screen placement of UQLAB.
2 Description
uq_figure wraps the MATLAB figure command and sets its formatting style following
UQLAB defaults. By default, it also places a newly created figure in the center of the screen.
The input arguments to uq_figure (and most of their default values) as well as the result-
ing figure behavior are consistent with the figure command. A list of default properties
specifically set by uq_figure is available in Table 1.
Table 1: uq_figure defaults
'Color' 'white'
'Position' centered (depends on the actual screen size) with a width and
height of 798 and 600 pixels, respectively.
'Renderer' 'opengl'
'Name' 'uq_figure_(figure number)' (the figure number uses
an internal counter)
'Filename' 'uq_figure_(figure number).fig' (the figure number
uses an internal counter)
3 Syntax
uq_figure
uq_figure(...)
uq_figure(..., NAME, VALUE)
F = uq_figure(...)
uq_figure creates a new formatted figure window.
uq_figure(...) wraps the MATLAB figure command, supporting the same input argu-
ments.
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uq_figure(..., Name, Value) also sets the properties of the figure according to the spec-
ified NAME/VALUE pairs.
F = uq_figure(...) returns the Figure object. Use F to access or modify the properties
of the figure after it is created. In MATLAB R2014a or older, F is the handle to the
Figure object.
4 Examples
4.1 Create a figure with default formatting
A new figure window following the default formatting of UQLAB can be opened using the
uq_figure command (see Figure 1(a) for the resulting figure):
uq_figure
4.2 Formatted figure
To modify the appearance of a figure window, the standard MATLAB name-value pairs can be
used. For example (see Figure 1(b) for the resulting figure):
uq_figure('Color', 'y', 'Name', 'myNewFigure5')
(a) Default formatting (b) Custom formatting
Figure 1: Figures with default and custom formatting.
5 Input
The input arguments for uq_figure are identical to those of the standard figure command
in MATLAB. Refer to its documentation for details.
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6 Output
Table 2: F = uq_figure(...)
F Figure object The object can be used to query or
modify the properties of the figure. In
MATLAB R2014a or older, F is a handle
(i.e., unique identifiers) to the Figure
object.
7 Notes
• In MATLAB R2014a or older, the function returns a handle to the Figure object. By
default, this figure handle has the same value as the figure number. In MATLAB R2014b
or newer, the function returns directly the Figure object.
UQLAB-V2.1-201 uq figure - 65 -
uq formatDefaultAxes – Default
formatting of an Axes object
1 Objective
Format an Axes object following UQLAB defaults.
2 Description
uq_formatDefaultAxes formats a given Axes object according to the UQLAB default for-
matting style given in Table 1.
Table 1: uq_formatDefaultAxes defaults (Axes object)
'Box' 'on'
'FontSize' Computed based on the size of the current figure
'Layer' 'top'
'LineWidth' 1.5
'TickLabelInterpreter' 'LaTeX'
'XGrid' 'on'
'YGrid' 'on'
'ZGrid' 'on'
Besides setting the properties of the Axes object, uq_formatDefaultAxes also sets the
default properties of several children Text objects: 'XLabel', 'YLabel', 'ZLabel', and
'Title'. The default properties for these objects are listed in Table 2.
Table 2: uq_formatDefaultAxes defaults (Text objects)
'FontSize' Computed based on the size of the current figure. This value is
identical to the one for the 'FontSize' property of the Axes
object.
'Interpreter' 'LaTeX'
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UQLIB – Graphics
3 Syntax
uq_formatDefaultAxes(AX)
uq_formatDefaultAxes(..., NAME, VALUE)
uq_formatDefaultAxes(ax) formats the Axes object AX according to the UQLAB default
formatting style.
uq_formatDefaultAxes(..., NAME, VALUE) uses NAME/VALUE pairs to override the de-
faults.
4 Examples
4.1 Create an Axes with default formatting
In the example below, a figure is created with an Axes which is then formatted by uq_formatDefaultAxes.
The result is shown in Figure 1(a).
uq_figure
uq_formatDefaultAxes(gca)
4.2 Create an Axes with custom formatting
To modify the Axes or override the defaults, a set of name-value pairs can be used. For
example:
uq_figure
uq_formatDefaultAxes(gca, 'XGrid', 'off', 'Box', 'off')
xlabel('time')
ylabel('$\mathcal{T}_j$')
will result in Figure 1(b).
(a) Default formatting (b) Custom formatting
Figure 1: Axes with default and custom formatting.
UQLAB-V2.1-201 uq formatDefaultAxes - 67 -
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5 Input
Table 3: uq_formatDefaultAxes(AX)
AX Axes object Target Axes.
NAME, VALUE name-value pair Additional options as name-value
pairs. They can be used to override
the defaults set according to Table 1.
6 Output
uq_formatDefaultAxes does not return any output.
7 Notes
• The Axes property 'TickLabelInterepreter' is not available in MATLAB R2014a or
older.
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uq plot – Create a formatted 2D line
plot
1 Objective
Create a 2-dimensional line plot following the default formatting of UQLAB.
2 Description
uq_plot wraps the MATLAB plot command and sets the default formatting style of UQLAB
on the resulting Figure and Axes objects. The input arguments of uq_plot conform to
the standard MATLAB plot function. The properties specifically set by uq_plot are given in
Table 1.
Table 1: uq_plot defaults
'LineWidth' 2.0
'MarkerFaceColor' Same as the line 'Color' property
3 Syntax
uq_plot(X,Y)
uq_plot(...)
uq_plot(..., NAME, VALUE)
uq_plot(AX,...)
H = uq_plot(...)
uq_plot(X,Y) plots vector Y versus vector X following UQLAB default formatting.
uq_plot(...) wraps the MATLAB plot function, supporting the same input arguments.
uq_plot(..., NAME, VALUE) modifies the properties of the plot according to the spec-
ified NAME/VALUE pairs. The complete list of NAME/VALUE pairs can be found in the
documentation of the MATLAB plot function.
uq_plot(AX,...) creates the plot into the Axes object AX instead of the current axes.
H = uq_plot(...) returns a column vector of Line objects. Use the elements in H to
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access modify the properties of specific plot elements after they are created. In MATLAB
R2014a or older, the function returns one or more handles to lineseries objects.
4 Examples
4.1 Create a line plot with default formatting
This example creates the 2-dimensional line plot following the default formatting of UQLAB.
The resulting plot is shown in Figure 1(a).
x = 0:pi/100:2
*
pi;
y = sin(x);
uq_plot(x,y)
4.2 Create a line plot with custom formatting
This examples creates the 2-dimensional line plot with custom formatting shown Figure 1(b).
x = 0:pi/10:2
*
pi;
y = sin(x);
uq_plot(...
x, y, '--s',...
'LineWidth', 2,...
'MarkerSize', 10,...
'MarkerEdgeColor', 'b',...
'MarkerFaceColor', [0.5 0.5 0.5])
(a) Default formatting (b) Custom formatting
Figure 1: A 2-dimensional line plot with default and custom formatting.
Note that all the valid input arguments are consistent with the plot function in MATLAB.
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5 Input
The input arguments for uq_plot are identical to the ones for the MATLAB’s plot function.
Refer to its documentation for details.
6 Output
Table 2: H = uq_plot(...)
H Scalar or vector of Line
Objects
Use the object to access and modify the
properties of a specific line. In MATLAB
R2014a or older, they are unique
identifiers (i.e., handles) to
lineseries objects.
7 Note
• Prior to plotting, uq_plot searches for the current active figure. If none is found, a
new figure window is created using uq_figure. It then searches for an Axes object. If
none is found, a new axes is created and formatted according to the UQLAB defaults
by uq_formatDefaultAxes.
• In MATLAB R2014a or older, the function returns one or more handles to lineseries
object. In MATLAB R2014b or newer, the function returns one or more Line objects.
• uq_plot function wraps the MATLAB plot function and it accepts all the input argu-
ments of the MATLAB function. The default of uq_plot can be overridden by passing
other values to the default properties (Table 1) during the function call.
UQLAB-V2.1-201 uq plot - 71 -
uq bar – Create a bar plot
1 Objective
Create a bar graph following the default formatting of UQLAB.
2 Description
uq_bar wraps the MATLAB bar function and sets the default formatting style of UQLAB on
the resulting Figure and Axes objects. The input arguments to uq_bar follow those of the
default bar function in MATLAB. The properties specifically set by uq_bar can be found in
Table 1.
Table 1: uq_bar defaults
'EdgeColor' 'none'
3 Syntax
uq_bar(Y)
uq_bar(X,Y)
uq_bar(..., NAME, VALUE)
uq_bar(AX,...)
uq_bar(...)
H = uq_bar(...)
uq_bar(Y) creates a set of bar graphs, one for each element in Y, following the default
formatting of UQLAB.
uq_bar(X,Y) creates a set of bar graphs, one for column of Y at the locations specified in X.
uq_bar(...) wraps the MATLAB bar function, supporting the same input arguments.
uq_bar(..., NAME, VALUE) modifies the properties of the plot according to the specified
NAME/VALUE pairs. The complete list of the available NAME/VALUE pairs can be found
in the documentation of the MATLAB bar function.
uq_bar(AX,...) creates the plot into the axes AX instead of the current axes.
H = uq_bar(...) returns one or more Bar objects. Use the elements in H to access and
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modify the properties of a specific Bar object after it has been created. In MATLAB
R2014a or older, the function returns one or more handles to barseries objects.
4 Examples
4.1 Create a bar plot with default formatting
A simple bar plot with the UQLAB default formatting style is created in the example below.
The resulting plot is shown in Figure 1(a).
y = [90 81 155 113.5 121 140 129 201 216 259 231.5];
uq_bar(y)
4.2 Create a bar plot with custom formatting
A bar plot can be further customized as illustrated in the example below. The resulting plot
is shown in Figure 1(b).
y = [90 81 155 113.5 121 140 129 201 216 259 231.5];
uq_bar(y,...
'FaceColor', [0.5 0.35 0.5],...
'EdgeColor',[0.1 0.1 0.1],...
'LineWidth', 2.5)
Note that all the valid input arguments are consistent with MATLAB bar function.
(a) Default formatting (b) Custom formatting
Figure 1: A bar plot with default and custom formatting style.
4.3 Create a bar plot with different bar locations
The location of the bars can be specified as illustrated with the example below. The resulting
plot is shown in Figure 2.
UQLAB-V2.1-201 uq bar - 73 -
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x = 2014:1:2019;
y = [75 91 105 123.5 131 281.5];
uq_bar(x,y)
Figure 2: A bar plot with custom locations of the bars.
5 Input
The input arguments for uq_bar are identical to the ones for the MATLAB’s bar function.
Refer to its documentation for details.
6 Output
Table 2: H = uq_bar(...)
H Scalar or vector of Bar
Objects
Use the object to access and modify the
properties of a specific bar plot. In
MATLAB R2014a or older, they are
handles (i.e., unique identifiers) to
barseries objects.
7 Notes
• In MATLAB R2014a or older, the function returns one or more handles to barseries
objects. In MATLAB R2014b or newer, the function returns one or more Bar objects.
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uq histogram – Create a histogram
1 Objective
Create a histogram using the default formatting of UQLAB.
2 Description
Given a set of univariate data points, uq_histogram creates a histogram, a type of bar plot
that groups the data into bins. By default, the resulting histogram is normalized, that is, the
area under the histogram is summed up to 1.0. uq_histogram uses an automatic binning
algorithm that returns bins with uniform width. The width of the bins are computed by
Scott’s rule (Scott, 2010)
w = 3.49 ˆσ
y
n
−1/3
(1)
where w is the width of the bins; n is the number of data points; and ˆσ
y
is the empirical
standard deviation of the data points.
3 Syntax
uq_histogram(Y)
uq_histogram(X,Y)
uq_histogram(..., 'Normalized', false)
uq_histogram(..., NAME, VALUE)
uq_histogram(AX,...)
H = uq_histogram(...)
[H,N] = uq_histogram(...)
[H,N,X] = uq_histogram(...)
uq_histogram(Y) plots a histogram of a column- or row-vector Y with normalized values
(i.e., total area under the histogram is 1.0). uq_histogram uses an automatic binning
algorithm that returns bins with a uniform width.
uq_histogram(X,Y) plots a histogram of Y among bins with the centers specified by vector
X.
uq_histogram(..., 'Normalized', false) plots an unnormalized (i.e., raw counts)
histogram. By default, the value of the named argument 'Normalized' is true.
uq_histogram(..., NAME, VALUE) modifies the properties of the plot according to the
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specified NAME/VALUE pairs. The complete list of the available NAME/VALUE can be
found in the documentation of the MATLAB bar function.
uq_histogram(AX,...) creates the plot into the Axes object AX instead of the current axes.
H = uq_histogram(...) returns the underlying Bar object used to create the plots. Use
H to access and modify the properties of the underlying Bar object after it has been
created. In MATLAB R2014a or older, the function returns the handle to barseries
object.
[H,N] = uq_histogram(...) additionally returns the number of elements per bins (con-
tainers). The values might be normalized depending on the option set by the named
argument 'Normalized'.
[H,N,X] = uq_histogram(...) additionally returns the center position of the bins along
the x-axis.
4 Examples
4.1 Create a default histogram
A simple histogram plot following the default formatting of UQLAB is created in the example
below. The resulting plot is shown in Figure 1(a).
y = randn(1000,1);
uq_histogram(y)
4.2 Create a customized histogram
A histogram created by uq_histogram can be further customized as illustrated in the exam-
ple below. The resulting plot is shown in Figure 1(b).
y = randn(1000,1);
uq_histogram(...
y,...
'FaceColor', [0.5 0.35 0.5],...
'EdgeColor',[0.1 0.1 0.5],...
'LineWidth', 2.5)
4.3 Create multiple histograms
Multiple histograms can be created in the same figure as illustrated in the example below in
the case of two histograms. The resulting plot is shown in Figure 2.
y1 = randn(1000,1);
y2 = 2 + 2
*
randn(1000,1);
uq_histogram(y1)
hold on
uq_histogram(y2, 'FaceAlpha', 0.8)
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(a) Default formatting (b) Custom formatting
Figure 1: A histogram with default and custom formatting style.
Figure 2: Multiple histograms.
4.4 Create a raw count histogram
By default, the histogram created by uq_histogram is normalized such that the area under
the histogram is equal to 1.0. To create a histogram with raw counts, use the example below
with the resulting plot shown in Figure 3.
y = randn(1000,1);
uq_histogram(y, 'Normalized', false)
UQLAB-V2.1-201 uq histogram - 77 -
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Figure 3: A histogram with raw counts.
5 Input
Table 1: uq_histogram(...)
Y 1 × N or N × 1 Double Vector of input data
X 1 × K or K × 1 Double Vector of bin centers along the
x−axis.
AX Axes object Axes object in which the histogram
will be created. If not specified,
uq_histogram uses the current
axes for the histogram.
NAME, VALUE name-value pair (See
Table 2)
Additional options as name-value
pairs.
Table 2: uq_histogram(..., NAME, VALUE)
'Normalized' Logical
default: true
Histogram normalization. If
normalized, the total area of the
histogram equals 1.0.
'ColorRange' 1 × 2 Double A pair that defines the range used to
color the histogram bars. The bars in
the histogram are colored according to
their height by linearly interpolating
between the specified color pair. This
option is only supported in MATLAB
R2017b or newer; In older MATLAB, the
option is ignored.
Note: A histogram constructed by uq_histogram consists of bar elements created by
uq_bar. Therefore, all the name-value pairs valid for uq_bar are also valid for
uq_histogram .
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6 Output
Table 3: [H,N,X] = uq_histogram(...)
H Bar Object Use the object to access and modify the
properties of the histogram. In MATLAB
R2014a or older, it is the handle (i.e.,
unique identifier) to a barseries
object.
N Vector of Double Number of elements (data points) per
bins. The values might be normalized
depending on the value of the
'Normalized' option.
X Vector of Double Center positions of the bins along the
x-axis.
7 Notes
• In MATLAB R2014a or older, the function returns the handle to a barseries object. In
MATLAB R2014b or newer, the function returns a Bar object.
UQLAB-V2.1-201 uq histogram - 79 -
uq violinplot – Create violin plots
1 Objective
Create violin plots.
2 Description
A violin plot is used to visualize the distribution of a data set. The plot is similar in principle
to a Tuckey box-plot, but instead only showing interquantile ranges it shows kernel-density
estimates of the data density. The plot is especially useful to visualize multimodal and highly
skewed distributions.
3 Syntax
uq_violinplot(Y)
uq_violinplot(X,Y)
uq_violinplot(..., NAME, VALUE)
uq_violinplot(AX,...)
H = uq_violinplot(...)
uq_violinplot(Y) generates M a violin plot based on the N-by-M data matrix Y.
uq_violinplot(X,Y) generates violin plots centered on specified points in X. The number
of elements in X must be consistent with the number of columns in Y.
uq_violinplot(..., NAME, VALUE) modifies the properties of the plot according to the
specified NAME/VALUE pairs. The complete list of NAME/VALUE pairs can be found in
the documentation of the MATLAB patch function.
uq_violinplot(AX,...) creates the plot into the axes AX instead of the current axes.
H = uq_violinplot(...) returns one or more Patch objects. Use the elements in H to
access and modify the properties of a specific violin plot. In MATLAB R2014a or older,
the function returns one or more handles to patch objects.
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UQLIB – Graphics
4 Examples
4.1 Create a violin plot with default formatting
A series of violin plots following the default formatting style of UQLAB is created in the
example below. The resulting plot is shown in Figure 1(a).
y = [1 2 3] + [0.5 1.0 2] .
*
randn(1000,3);
uq_violinplot(y)
4.2 Create a violin plot with custom formatting
A series of violin plots can be further customized as illustrated in the example below. The
resulting plot is shown in Figure 1(b).
y = [1 2 3] + [0.5 1.0 2] .
*
randn(1000,3);
uq_violinplot(...
y,...
'FaceColor', [0.5 0.35 0.5],...
'EdgeColor',[0.1 0.1 0.5],...
'LineWidth', 2.5)
(a) Default formatting (b) Custom formatting
Figure 1: Violin plots with default and custom formatting style.
4.3 Create multiple violin plots
Multiple series of violin plots can be created as illustrated in the example below. The resulting
plot is shown in Figure 2.
rng(100,'twister') % f o r r e p r o d u c i b i l i t y
y1 = [1 2 3] + betarnd(0.5, 0.5, [1000 3]); % Bimo da l b e t a
y2 = [1 2 3] + [0.15 0.25 0.30] .
*
randn(1000,3); % G au s si a n
cmap = uq_colorOrder(2);
uq_violinplot(y1)
hold on
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uq_violinplot(y2, 'FaceColor', cmap(2,:), 'FaceAlpha', 0.8)
hold off
Figure 2: A bar plot with custom locations of the bars.
5 Input
Table 1: uq_violinplot(...)
Y N × M Double Data matrix, M violin plots are
created for each column of Y.
X 1 × M or M × 1 Double The center positions of each violin
plot along the x-axis.
AX Axes object Axes object in which the violin plots
will be created. If not specified, the
uq_violinplot uses the current
axes for the plot.
NAME, VALUE name-value pair (See
Table 2)
Additional options as name-value
pairs.
Table 2: uq_violinplot(..., NAME, VALUE)
'FaceColor' 1 × 3 Double
default:
uq_colorOrder(1)
Vector specifying the filling color of the
violin plot. The color is specified in the
RGB scale.
'EdgeColor' 1 × 3 Double or String
default: 'none'
Vector specifying the edge color of the
violin plots. The color is specified in the
RGB scale.
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Note: A violin plot consists of patch elements created by MATLAB patch func-
tion. Therefore, all the name-value pairs valid for patch are also valid for
uq_violinplot .
6 Output
Table 3: H = uq_violinplot(...)
H Scalar or vector of Patch
objects
Use the object to access and modify the
properties of a specific violin plot. In
MATLAB R2014a or older, they are
handles (i.e., unique identifiers) to
patch objects.
7 Notes
• In MATLAB R2014a or older, the function returns one or more handles to patch objects.
In MATLAB R2014b or newer, the function returns one or more Patch objects.
• The 'FaceColor' property of violin plots is not cycled through using the hold on
statement. Plotting multiple violin plots on the same figure, therefore, require manually
setting their color.
UQLAB-V2.1-201 uq violinplot - 83 -
uq scatterDensity – Create a
scatter plot matrix
1 Objective
Create a scatter plot matrix of multivariate data.
2 Description
Given an N ×M multivariate data set X, uq_scatterDensity creates a scatter plot matrix,
a collection of M ×M plots organized into a matrix. The diagonal elements of the scatter plot
matrix visualize the histograms of each column of the input data. The (lower) off-diagonal
elements of the plot visualizes the pairwise scatter plots of the corresponding columns of X.
The (upper) off-diagonal elements of a scatter plot matrix created by uq_scatterDensity
are empty.
3 Syntax
uq_scatterDensity(X)
uq_scatterDensity(..., hist_NAME, VALUE, scatter_NAME, VALUE)
uq_scatterDensity(..., NAME, VALUE)
uq_scatterDensity(AX,...)
H = uq_scatterDensity(...)
[H,AXS] = uq_scatterDensity(...)
[H,AXS,BIGAX] = uq_scatterDensity(...)
uq_scatterDensity(X) creates a scatter plot matrix from each pairs of columns in X. If
X is N-by-M matrix, uq_scatterDensity produces a lower triangular matrix of M-by-M
plots, in which histograms of each column of X are plotted in the main diagonal and
scatter plots of each pair of columns of X are plotted as the off-diagonal elements.
uq_scatterDensity(...,hist_NAME, VALUE, scatter_NAME, VALUE) modifies the scat-
ter plot matrix using additional NAME/VALUE pair arguments. The following convention
applies when providing pairs to the corresponding elements of the matrix of plots:
– If NAME has the prefix 'hist_', then VALUE is passed to the function uq_histogram
to create the histograms in the main diagonal;
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UQLIB – Graphics
– If NAME has the prefix 'scatter_', then VALUE is passed instead to the uq_plot
function that creates the off-diagonal scatter plots.
uq_scatterDensity(..., NAME, VALUE) modifies the overall plot using NAME/VALUE pair
arguments of which the NAMEs have neither the prefix 'hist_' nor 'scatter_'.
uq_scatterDensity(AX,...) creates the plot into the Axes object AX instead of the cur-
rent axes.
H = uq_scatterDensity(...) returns the graphics objects of the plot. The main diagonal
elements (histograms) are represented by Bar objects and the off-diagonal elements are
represented by Line objects. Use the elements in H to access and modify the properties
of specific plots. In MATLAB R2014a or older, the function returns handles to barseries
objects (instead of Bar) and lineseries objects (instead of Line).
[H,AXS] = uq_scatterDensity(...) additionally returns the Axes objects (or handles)
of all the plots inside the scatter plot matrix.
[H,AXS,BIGAX] = uq_scatterDensity(...) additionally returns the parent axes of the
scatter plot matrix.
4 Examples
4.1 Create a scatter plot matrix with default formatting
A scatter plot matrix following the default formatting style of UQLAB is created in the exam-
ple below. The resulting plot is shown in Figure 1.
rng(100,'twister') % f o r r e p r o d u c i b i l i t y
x = mvnrnd([1 -1 3], [0.9 0.4 0.0; 0.4 0.3 0.0; 0.0 0.0 1.0], 1000);
uq_scatterDensity(x)
4.2 Create a scatter plot matrix with custom formatting
A scatter plot matrix can be further customized as illustrated in the example below. The
resulting plot is shown in Figure 2.
rng(100,'twister') % f o r r e p r o d u c i b i l i t y
x = mvnrnd([1 -1 3], [0.9 0.4 0.0; 0.4 0.3 0.0; 0.0 0.0 1.0], 1000);
uq_scatterDensity(...
x,...
'hist_FaceColor', [0.5 0.35 0.5],...
'hist_EdgeColor', 'k',...
'scatter_MarkerEdgeColor', [0.9020 0.3333 0.0510],...
'labels', {'$X_1$', '$X_2$', '$X_3$'})
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Figure 1: A scatter plot matrix with default formatting style.
Figure 2: A scatter plot matrix with custom formatting style.
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5 Input
Table 1: uq_scatterDensity(...)
X N × M Double Input data matrix
AX Axes object Axes object in which the violin plots
will be created. If not specified,
uq_scatterDensity uses the
current axes for the plot.
hist_NAME,
hist_VALUE
name-value pair Additional options as name-value
pairs for the histograms in the main
diagonal (see uq_histogram).
scatter_NAME,
scatter_VALUE
name-value pair Additional options as name-value
pairs for the scatter plots in the
off-diagonal (see uq_plot).
NAME, VALUE name-value pair (See
Table 2)
Additional options as name-value
pairs
Table 2: uq_scatterDensity(..., NAME, VALUE)
'points' N
1
× M Double A set of N
1
points that are plotted
together with X in the scatter plots
'labels' 1 × M Cell Array Labels for the subplots
'color' 1 × 3 Double Vector specifying the color of the plotted
elements. The color is specified in the
RGB scale.
'title' String Title string to add to the scatter plot
matrix
6 Output
Table 3: [H,AXS,BIGAX] = uq_scatterDensity(...)
H M × M objects array Graphics objects of the plots: Bar and
Line objects for the histograms and
scatter plots, respectively. In MATLAB
R2014a or older, they are handles (i.e.,
unique identifiers) to barseries and
lineseries objects.
AXS M × M Axes objects
array
Axes objects (or handles) of all plots
inside the scatter plot matrix
BIGAX Axes Object Parent axes of the scatter plot matrix
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uq traceplot – Create trace plots
1 Objective
Create trace plots of series of points with kernel density-based marginal estimates .
2 Description
Given a matrix of N × M random realizations X, trace plots show the sampled values over
consecutive realizations for each column of X. For each trace plot, a kernel density estimate
plot of all realizations is also plotted side by side. Trace plots are typically used in the visual
assessment of Markov Chain Monte Carlo simulation to inspect the behavior and convergence
of the chain (see also UQLAB User Manual – Bayesian inference for model calibration and
inverse problems).
3 Syntax
uq_traceplot(X)
uq_traceplot(..., NAME, VALUE)
uq_traceplot(AXES,...)
PLOTAX = uq_traceplot(...)
uq_traceplot(X) plots the trace plots of an N-by-M-by-C array X. By default, M figures are
created.
uq_traceplot(..., NAME, VALUE) modifies the properties of the trace plot according to
the specified NAME/VALUE pair.
uq_traceplot(AXES,...) plots into a cell array of Axes objects AXES.
PLOTAX = uq_traceplot(....) returns the Axes objects of the generated plot in an M-by-
2 cell array. In MATLAB R2014a or older, the function returns a cell array with handles
to the Axes objects.
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4 Examples
4.1 Create trace plots
Two trace plots based on 500 realizations of a bivariate normal distribution are created fol-
lowing the default formatting of UQLAB in the example below.
rng(100,'twister') % f o r r e p r o d u c i b i l i t y
x = mvnrnd([1 -1], [0.9 0.4; 0.4 0.3], 500);
uq_traceplot(x)
The resulting plot is shown in Figure 1.
Figure 1: Two trace plots based on a 500 × 2 matrix.
5 Input
Table 1: uq_traceplot(...)
X N × M × C Double Data matrix. If a 3-dimensional array
is given, then for each trace plot, C
separate lines are created.
AXES M × 2 Cell array of Axes
objects
Axes objects in which the trace plots
will be created
NAME, VALUE name-value pair (See
Table 2)
Additional options as name-value
pairs
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Table 2: uq_traceplot(..., NAME, VALUE)
'labels' 1 × M Cell Array
default:
{'X_1','X_2',...,'X_M'}
Cell array of strings with labels used in
the plots
6 Output
Table 3: PLOTAX = uq_traceplot(...)
PLOTAX M × 2 Cell array of Axes
objects
Axes objects in which the trace plots
are created. In MATLAB R2014a or
older, they are handles (i.e., unique
identifiers) to the Axes objects. The
first and second columns correspond to
the trace and kernel density plots for
each data dimension, respectively.
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uq legend – Create a legend
1 Objective
Create a legend following the default formatting of UQLAB.
2 Description
uq_legend wraps the legend function in MATLAB and sets the UQLAB default formatting
style on the resulting Legend object. The input arguments of uq_legend conform to the
arguments of legend. The properties specifically set by uq_legend are listed in Table 1.
Table 1: uq_legend defaults
'Box' 'on'
'Color' 'white'
'EdgeColor' 'white'
'Interpreter' 'LaTeX'
'Location' 'best'
3 Syntax
uq_legend
uq_legend(...)
uq_legend(..., NAME, VALUE)
LGD = uq_legend(...)
uq_legend creates a legend on the current axes following the default formatting of UQLAB.
uq_legend(...) wraps the standard MATLABlegend function and sets the default format-
ting of UQLAB on the resulting Legend object. The input arguments to uq_legend
conform to the input arguments of the MATLAB function.
uq_legend(..., NAME, VALUE) modifies the properties of the legend according to the
specified NAME/VALUE pairs. The complete list of the pairs can be found in the docu-
mentation of the legend function.
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LGD = uq_legend(...) returns the Legend object. In MATLAB R2014a or older, the func-
tion returns a handle to the legend (which is an Axes object). Use LGD to access and
modify the properties of the legend after it has been created.
4 Examples
4.1 Add a legend to a plot with the default formatting
Below is an example how to create a legend with uq_legend. The resulting legend is shown
in Figure 1(a).
x = repmat(linspace(0,1)',1,3);
y = x .
*
(1:3);
uq_plot(x,y)
uq_legend('Data 1', 'Data 2', 'Data 3')
4.2 Add a legend to a plot with a custom formatting
To customize the legend, a set of name-value pairs can be used. The default values listed in
Table 1 can be overriden using the same approach. Below is an example of customizing a
legend created using uq_legend. The resulting legend is shown in Figure 1(b).
x = repmat(linspace(0,1)',1,3);
y = x .
*
(1:3);
uq_plot(x,y)
uq_legend(...
{'Data 1', 'Data 2', 'Data 3'},...
'Box', 'on',...
'EdgeColor', 'r',...
'Location', 'southeast',...
'FontSize', 20)
(a) Default formatting (b) Custom formatting
Figure 1: Two plots with default and custom formatting styles for the legends.
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5 Input
The input arguments for uq_legend are identical to the ones of the legend function in
MATLAB. Refer to its documentation for details.
6 Output
Table 2: H = uq_legend(...)
LGD Legend Object Use the object to access and modify the
properties of the legend. In MATLAB
R2014a or older, the function returns a
handle (i.e., unique identifiers) to the
legend, which is also an Axes.
7 Notes
• In MATLAB R2014a or older, the 'FontSize' property of the legend has the same value
as the 'FontSize' property of the parent Axes object. In MATLAB R2014b or newer,
the 'FontSize' property of the legend is, by default, 90% of the the 'FontSize'
property of the parent Axes object.
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uq map – Map a sequence using a
function
1 Objective
Evaluate a function on each element of an input sequence and return the evaluation results
inside a cell array of the same length as the length of the input sequence.
2 Description
Given an input sequence and a function, uq_map evaluates the given function on each el-
ement of the sequence. In other words, uq_map defines a mapping from one sequence to
another sequence using the given function (from here on in refered to as a mapping func-
tion). uq_map always returns a cell array of the same length as the input sequence; each
element of the cell array contains the results of the function evaluations. In effect, the results
of each function evaluation may have different data types. Figure 1 illustrates the process of
mapping a sequence to another by uq_map.
Figure 1: An illustration of a mapping process by uq_map.
Note: uq_map is a dispatcher-aware function. That is, the function is readily dispatch-
able to a remote machine and, when possible, executed in parallel using a DIS-
PATCHER object. For details on using uq_map coupled with a DISPATCHER object,
the reader is referred to the UQLAB User Manual – The HPC Dispatcher module,
in particular, Appendix C (Wicaksono et al., 2021).
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2.1 Types of mapping function
uq_map supports different types of mapping functions as its input, including MATLAB built-
in functions, user-defined functions, anonymous functions, and system commands. Table 1
provides some remarks for each of these types.
Table 1: Supported functions for an input to uq_map.
Mapping function type Remark
MATLAB built-in functions All built-in functions are supported.
User-defined functions The function m-file (and all its dependencies) must be avail-
able in the MATLAB path.
Anonymous functions If an anonymous function is assigned to a variable, the vari-
able is passed to uq_map without the preceding @.
System command
• System commands are specified as a string (i.e., char
array) and executed using system command of MAT-
LAB.
• Each mapping function evalution returns a struct that
captures the exit status and terminal output from the
command execution.
• Valid system commands are specific to the operating
system (OS).
The mapping function, with the exception of system command, is passed to uq_map as a
function handle.
2.2 Types of sequence
uq_map takes a sequence represented by different MATLAB data types, including structure
arrays, cell arrays, vectors, and matrices. The notion of an element of the sequence differs
from type to type, and with an optional named argument, it can be modified as explained
below.
Structure arrays are the most straightforward type of sequence to map using uq_map. This
is illustrated in Figure 2 where each element of the structure array is used as input to the
mapping function. Notice that the shape of the input structure array is preserved in the
output, including in higher dimensions.
By default, when the input sequence is a cell array, taking an element out of it behaves the
same way as in the case of structure array. The content (i.e., value) of cell array element is
passed to the mapping function. Being a cell array, the content of an element can be of any,
even mixed, types.
However, with named argument 'ExpandCell' followed by a logical true or false, this
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Figure 2: An illustration of a mapping process by uq_map on a structure array.
behavior can be changed. If 'ExpandCell' is set to true, then the content of the cell
array element is expanded into a comma-separated list. This construction is useful if the
input sequence has as its element an argument list for the mapping function. The difference
between two settings of 'ExpandCell' is illustrated in Figure 3. Note once more that the
shape of the input cell array is preserved in the output cell array.
Figure 3: An illustration of a mapping process by uq_map on a cell array.
Note: The named argument 'ExpandCell' is only relevant for a cell array as the input
sequence uq_map.
When the input sequence is a numerical vector (or a matrix), then by default, each element
of the vector (resp, matrix) is passed to the mapping function. This behavior can be changed
using the named argument 'MatrixMapping' and it is particularly relevant if the input is a
matrix. Three different values are possible:
• 'ByElements': Each element of the vector or matrix, i.e., a scalar, is passed to the
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mapping function (the default).
• 'ByRows': Each row of the vector or matrix, i.e., a row vector, is passed to the mapping
function. If the input sequence is a row vector, then the sequence consists of only one
element.
• 'ByColumns': Each column of the vector or matrix, i.e., a column vector, is passed
to the mapping function. If the input sequence is a column vector, then the sequence
consists of only one element.
Figure 4 illustrates the different ways of taking an element from a matrix.
Figure 4: An illustration of a mapping process by uq_map on a matrix.
Note: uq_map only supports numerical array up to 2 dimension (i.e., a matrix). In this
case, the input element would either be a scalar or a vector. To use an array of
higher dimension as an element passed to the mapping function, use cell arrays
instead.
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2.3 Passing parameters
A argument of the mapping function may be passed as a parameter. While the mapping func-
tion is evaluated on each element of the input sequence, the value of a parameter argument
will be kept as-is in each evaluation. An argument passed as a parameter is always positioned
the last in the argument list of the mapping function. Figure 5 illustrates calling uq_map with
a parameter.
Figure 5: An illustration of a mapping process with a parameter.
A parameter is specified in uq_map using the named argument 'Parameters' followed by the
parameter. The parameter can take any MATLAB data types as long as it is a valid argument
for the mapping function.
2.4 Mapping functions with multiple outputs
uq_map supports evaluating mapping functions that return multiple output arguments. If a
mapping function returns multiple output arguments, uq_map can be called with multiple
output arguments as well. The results of uq_map will then be N
out
cell arrays, where N
out
is
the number of requested output arguments. These cell arrays contain the combined outputs
from the mapping function evaluation on each element of the input sequence.
An example is provided in Section 4.5 to illustrate calling uq_map on a mapping function
with multiple outputs.
2.5 Handling errors
By default, any error from evaluating a mapping function is automatically handled by re-
turning NaN values; whose dimension is consistent with the specified number of output ar-
guments. For a diagnostic purpose, this can be turned off by using the named argument
'ErrorHandler' followed by a logical false. If error handling is turned off, any error
thrown from a mapping function evaluation will causes uq_map to fail.
Moreover, a custom user-defined error handler may also be specified. If a mapping function
throws an error, then the user-defined error handler catches the error and takes the steps
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specified in the handler. The error handler either must throw an error or return the same
number of outputs as specified for the mapping function (in most cases varargout will
suffice).
The user-defined error handle is a function, specified as a comma-separated pair consisting
of 'ErrorHandler' and a function handle. If the mapping function throws an error, then the
error handler specified by 'ErrorHandler' catches the error and takes the action specified
in the function.
A user-defined error handler has the following signature:
function varargout = myErrorHandler(S,varargin)
The first input argument of the error handler is a structure with these fields:
• identifier: Error identifier
• message: Error message text
• index: linear index into the input sequence at which the mapping function threw the
error
The remaining input arguments (a cell array) to the error handler are the input arguments
for the call to the mapping function that made the function throws the error.
When an error is thrown, the error handler is called. Within this function, users then have ac-
cess to the error identifier and message as well as the index of the sequence and the complete
argument list that cause the error. It is up to the users to return this information for diag-
nostic purposes or some selected values to replace the return value of the mapping function
return in the case of error.
2.6 System commands as a mapping function
Using system commands as a mapping function is a special use case of uq_map. In this case, a
system command (represented as char array) is executed by MATLAB system command. The
results of executing system, namely the command exit status and the command terminal
output, are returned by uq_map.
The system command itself is typically composed from a template. In a command template,
replacement fields are used as placeholders that later on will be replaced with the actual
values from the input sequence; the resulting char array is then passed to MATLAB system.
When using system command as a mapping function, only 2-level nested cell arrays are
supported as the input sequence. Each cell array element contains a list of values used to
replace the placeholders in the command template.
A replacement field has the following syntax:
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{<posID>[:<formatSpec>]}
The field, delimited by curly braces, contains two components:
1. <posID> specifies the position ID of the value from the cell array. Using position ID, a
scalar integer, values from the input can be flexibly arranged.
2. <formatSpec> (optional) specifies how the value should be printed. It uses the stan-
dard MATLAB formatting operators. If not specified, the formatSpec for character
vector (%s) is used.
Figure 6 provides an illustration how the replacement field works in practice when system
command is specified as the mapping function to uq_map. The illustration shows how a single
command works, but there is no limitation of how many commands are to be executed as
long as the command char is composed properly.
Figure 6: An illustration of a mapping process using a system command as the mapping
function.
Note: While uq_map returns the exit status and the terminal output from executing the
command, they are seldom the main purpose of executing the command. With
respect to the user’s MATLAB session, the side-effects (e.g., executing external
codes) of executing such command are often the main goal.
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3 Syntax
OUTPUT = uq_map(FUN,INPUTS)
[OUTPUT_1,...,OUTPUT_NOUT] = uq_map(...)
[...] = uq_map(..., NAME, VALUE)
OUTPUT = uq_map(FUN,INPUTS) maps a sequence INPUTS to another sequence OUTPUT by
evaluating a function handle FUN on each element of INPUTS. OUTPUT is a cell array
with the number of elements the same as in INPUTS. If any evaluation of FUN on the
elements of INPUTS fails, a NaN is returned.
[OUTPUT_1,...OUTPUT_NOUT] = uq_map(FUN,INPUTS) maps a sequence INPUTS using
FUN to a set of cell arrays OUTPUT_1,...,OUTPUT_NOUT where NOUT is the number of
requested outputs from FUN.
[...] = uq_map(..., NAME, VALUE) maps INPUTS by evaluating FUN on each of its ele-
ments with additional (optional) NAME/VALUE pairs (see Table 3).
4 Examples
4.1 Map a structure array
Create a structure array that contains random numbers:
rng(100,'twister') % f o r r e p r o d u c i b i l i t y
for i = 1:4
S(i).X = randn(1e3,1);
S(i).Y = 1 + 2
*
randn(1e3,1);
end
Compute the mean of field X and field Y for each element of the structure array:
output = uq_map(@(s) [mean(s.X) mean(s.Y)],S)
output =
1x4 cell array
{1x2 double} {1x2 double} {1x2 double} {1x2 double}
A custom function handle is created to combine the results of the MATLAB built-in functions
mean of each field.
The result of the first element of the structure array is indeed a row vector:
output{1}
ans =
-0.0192 0.9348
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Notice that the size of the input array is the same as the output array.
4.2 Map a cell array
Create an anonymous function that creates a sample of 1000 points from normal distribution
with parametrized mean and standard deviation:
rng(100,'twister') % F or r e p r o d u c i b i l i t y
fun = @(x) x{1} + x{2}
*
randn(1e3,1);
Create a cell array that contains the argument to the function handle:
C = {{0,1}; {1,2}; {2,3}};
Generate the samples from the three different arguments:
output = uq_map(fun,C)
output =
3x1 cell array
{1000x1 double}
{1000x1 double}
{1000x1 double}
Notice that if a handle is assigned to a variable, the variable name is passed directly to
uq_map.
Alternatively, a function handle may be defined as follows:
fun = @(mu,sd) mu + sd
*
randn(1e3,1);
In this case, the element of the input cell array can be directly expanded to a comma-
separated list and passed as an argument list into the mapping function:
output = uq_map(fun, C, 'ExpandCell', true)
output =
3x1 cell array
{1000x1 double}
{1000x1 double}
{1000x1 double}
The results are identical. To verify the samples, compute the mean:
uq_map(@mean,C)
ans =
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3x1 cell array
{[0.0604]}
{[0.9896]}
{[2.0411]}
and the standard deviation:
uq_map(@std,C)
ans =
3x1 cell array
{[1.0003]}
{[1.9727]}
{[2.8943]}
4.3 Map a matrix
Create a 4-by-3 random sample from a normal distribution:
A = rand(4,3);
Get the size of the each element using the size function:
output = uq_map(@size,A)
output =
4x3 cell array
{1x2 double} {1x2 double} {1x2 double}
{1x2 double} {1x2 double} {1x2 double}
{1x2 double} {1x2 double} {1x2 double}
{1x2 double} {1x2 double} {1x2 double}
By default, mapping a matrix uses each element of a matrix as an input to the mapping
function. Therefore, the output of size shows that the element is indeed a scalar:
output{1,1}
ans =
1 1
By using 'ByRows' for the named argument 'MatrixMapping', the elements will be taken
from the rows of the matrix:
uq_map(@size, A, 'MatrixMapping', 'ByRows')
output =
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4x1 cell array
{1x2 double}
{1x2 double}
{1x2 double}
{1x2 double}
The output of size shows that the element is a row vector:
output{1}
ans =
1 3
By using 'ByColumns' for the named argument 'MatrixMapping', the elements will be
taken from the columns of the matrix:
uq_map(@size, A, 'MatrixMapping', 'ByColumns')
output =
1x3 cell array
{1x2 double} {1x2 double} {1x2 double}
The output of size shows that the element is a column vector:
output{1}
ans =
4 1
4.4 Map a sequence using a parameter
A parameter can be passed to uq_map and, in turn, will be passed to each call to the mapping
function as a parameter; its value is the same for each call.
Create a cell array of 3 random matrices (each matrix has different number of rows):
for i = 1:3
C{i} = rand(i,1e3);
end
Compute the mean of each row of the matrices (the second argument of mean is set to 2):
uq_map(@mean, C, 'Parameters', 2)
output =
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3x1 cell array
{[ 0.5043]}
{2x1 double}
{3x1 double}
4.5 Map a sequence using a function with multiple outputs
Create a cell array of random matrices:
rng(100,'twister') % f o r r e p r o d u c i b i l i t y
for i = 1:3
C{i} = rand(5,5);
end
Compute the singular value decomposition (SVD) of each of these matrices using the svd
function (the svd function returns multiple outputs) and get all its outputs (i.e., the matrices
U, S, and V ):
[U,S,V] = uq_map(@svd, C)
U =
1x3 cell array
{5x5 double} {5x5 double} {5x5 double}
S =
1x3 cell array
{5x5 double} {5x5 double} {5x5 double}
V =
1x3 cell array
{5x5 double} {5x5 double} {5x5 double}
There are three output arrays that contain the combined outputs from the mapping function
evaluation on each element of the input sequence. For instance, the matrix stored in S{2} is
the matrix S from svd(C2):
S{1}
ans =
2.4977 0 0 0 0
0 1.0610 0 0 0
0 0 0.6075 0 0
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0 0 0 0.3753 0
0 0 0 0 0.3017
4.6 Handle an error
Create an anonymous function of two arguments:
fun = @(x,y) x + y;
Evaluate the function on a 2-element cell array using uq_map:
output = uq_map(fun,{1;2})
Because the function requires two inputs but each element of the sequence provides only
one, the evaluation fails. By default, a failed evaluation returns NaN:
output =
2x1 cell array
{[NaN]}
{[NaN]}
If the error handler is turned off, uq_map will throw an error instead:
output = uq_map(fun, {1;2}, 'ErrorHandler', false)
Not enough input arguments.
Error in @(x,y)x+y
...
A user-defined function handler can also be specified. The handler must be saved as a func-
tion in a separate m-file. For example:
function output = myErrorHandler(S,varargin)
output.ME = S;
output.Input = varargin;
end
The above error handler returns the error identifier and messages as well as the input ar-
gument on which the mapping function throws an error. These are useful for diagnostic
purposes.
Calling uq_map with the user-defined error handler:
output = uq_map(fun, {{1,2};{3,4}}, 'ErrorHandler', @myErrorHandler)
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output =
2x1 cell array
{1x1 struct}
{1x1 struct}
The details on the error for, for example, the first evaluation:
output{1}.ME
ans =
struct with fields:
identifier: 'MATLAB:minrhs'
message: 'Not enough input arguments.'
index: 1
4.7 Map a system command
Execute echo command (works both on Windows and Linux operating systems) on a set of
char arrays:
output = uq_map('echo {1}', {'Hello'; 'Goodbye'})
output1 =
2x1 cell array
{[0]}
{[0]}
output2 =
2x1 cell array
{'Hello' }
{'Goodbye'}
The exit status and the terminal output of the command execution are stored in the first and
second output, respectively.
5 Input
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Table 2: uq_map(FUN, INPUTS, NAME, VALUE)
FUN Function handle or char
array
Mapping function, used to evaluate
(i.e., map) each element of INPUTS.
See Table 1 for some remarks on the
supported types of function.
INPUTS Cell array, structure array,
or matrix
Sequence whose elements are passed
as inputs into FUN.
NAME, VALUE name-value pair (See
Table 3)
Additional options as name-value
pairs.
Table 3: uq_map(..., NAME, VALUE)
'Parameters' Any MATLAB data types
default: 'none'
Parameters passed to FUN as the last
positional argument. Parameters are
kept constant for each call to FUN. The
type of parameters depends on FUN.
See Section 2.3 for details.
'ExpandCell' logical
default: false
Flag to expand the content of a cell into
a comma-separated list. This is useful if
the contents of cell element are to be
pass as a list of arguments to FUN. See
Section 2.2 for an illustration.
'MatrixMapping' String
default: 'ByElements'
Way to take an element from a matrix.
Possible values:
• 'ByElements': each element is
an individual scalar from the
matrix.
• 'ByRows': each element is a
row vector corresponds to each
row of the matrix.
• 'ByColumns': each element is a
column vector corresponds to
each column of the matrix.
See Section 2.2 for an illustration.
Continued on next page
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UQLIB – Functional programming
Table 3–continued from previous page
'ErrorHandler' logical or function handle
default: true
Way to handle an error from evaluating
the mapping function on elements of
the input sequence. Possible values:
• true: Any error automatically
returns NaN.
• false: Errors are not handled;
any error causes uq_map to
throw an error.
• a function handle: a user-defined
error handler.
See Section 2.5 for details.
6 Output
Table 4: [OUTPUT_1,...,OUTPUT_NOUT] = uq_map(...)
OUTPUT_1,... Cell array(s) Results of evaluating FUN on each
element of INPUTS. The number of
elements in OUTPUT is always the same
as the number of elements in INPUTS.
If FUN supports multiple output
arguments, uq_map may also be called
with multiple output arguments. The
results are multiple cell arrays, each of
which contains each of the outputs from
FUN evaluation on the input sequence.
If a system command is used as FUN,
two output arrays may be requested.
Each of the output arrays has the same
size as INPUTS. The first and second
output array contains the exit status and
terminal output from executing the
command, respectively. (see
Section 2.6).
7 Notes
• uq_map is a dispatcher-aware function thus readily dispatchable to a remote machine.
However, if a DISPATCHER object is to be used, it is recommended to test uq_map locally
(i.e., without the DISPATCHER) on, possibly, limited set of the inputs to check if the call
is valid.
UQLAB-V2.1-201 uq map - 109 -
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